The question basically explains it all: I'm looking for the possibility and the means by which a creature can communicate with other members of its species with "telepathy".
I'm not looking for pheromone-emitting or just a silent "language" like a sign language, by the way.
Thinking about this rationally, you just need your creatures to communicate using a medium that isn't detected by other creatures in the environment.
This can be:
This is, to all intents and purposes, telepathy.
The informal definition of telepathy is effectively "by magic." The communication simply happens without any physics behind it.
But if you want to explore it a little more scientifically, then you have to define what you mean. I'm going to define it to mean "biological communication via a medium that's incompatible with the biology of outsiders." But there has to be a medium. If there's no medium, it's magic. Nothing wrong with that, lots of stories use it, but there's no point in trying to define "how." This also, by definition, means that the telepathy can be observed and understood by outsiders' application of a sufficient level of technology and effort. Everyone has the same laws of physics.
There are a variety of communications that aren't "telepathy" by this definition as they rely on observers simply not understanding the communication: jargon, code words, steganography, encryption, secret hand signs, shifting patterns of color, etc.
I'll ignore pheromones because the question said to, but they would be a decent option.
Touch-based sign language, based on applying varying forces and shapes of pressure would be an option. It's pretty mundane, as these things go, but hard to eavesdrop on. This is pretty close to what is already done by the community of people who are both blind and deaf. Our computers could learn it, given an appropriate interface device and a cooperative creature.
The "direct neural connection" idea in Jacob Badger's answer is a pretty good one. It would open up new disease vectors, and it obviously requires physical contact. On the downside, the bandwidth of a single nerve is pretty low, and a nerve bundle capable of carrying as much information as a conversation would be difficult to connect on demand. But it could happen.
UV light is another option. Many species, especially insects, can see light frequencies that humans can't. Bees use this to find nectar within flowers. A creature that had patches of UV-pigmented (or UV-emitting) skin and UV-sensitive vision could use this to communicate.
By the same token, ultrasonics could be a choice. Bats and dolphins only use it for echolocation, but they could use ultrasonics for communication too. 20th century technology could easily detect all this.
Going slightly more exotic, another option would be radio waves. Life on Earth never evolved sensitivity to radio, but maybe it could have, especially in an environment where sensing radio waves would be valuable. Some animals can create electrical currents, and electrical currents plus a conducting wire of the appropriate length makes a radio. There's iron in our blood (some animals use copper, which is even better). An organ could exist that maintains a radio antenna inside it which could be used to transmit and receive radio signals. 20th century level or more advanced humans would, of course, detect this quickly, but it would probably take a while to understand it. For a variation, you could instead use the electrical currents to modulate magnetic fields. The range would be short, and you only need Iron Age technology to detect it (but probably advanced computers to decode it).
Going more exotic than radio, you could try neutrinos. This is pushing the boundaries, but I don't think it's impossible. Neutrinos are difficult to detect and this communication might go unnoticed, at least for a while, by humans at a modern/near-future level of technology. The easiest way to produce neutrinos is through radioactive decay. A species whose biology was broadly compatible with radioactive elements could modulate the fission rate of the uranium in its body, and therefore its neutrino emissions, by increasing or decreasing the local concentration of it. Current state-of-the-art enables lightweight and compact neutrino detectors. A sufficiently large creature could contain a similar device, grown biologically. Nothing here is biologically impossible although there are a lot of stretches and dots to connect. Rate of information transfer would be low, but you can't beat the stealth.
More exotic than that? You're going into the realm of speculative or implausible physics. Gravitational waves are too difficult to generate by a plausible creature, and then you have undetected hypothetical particles, dark matter, etc. If the physics isn't understood or doesn't make sense for the purpose, you're into the realm of magic.
This would be otherwise impossible without the discovery of brain waves. Brain waves are electrical impulses in the brain.
So just make them be able to send out much more powerful brain waves that are easier to "pick up" in the form of thoughts and/or feelings. Then make your creatures also be able to receive and interpret those brain waves into the correct thought(s) and/or feeling(s) and bam! Telepathy.
Though they would only have so much of a range before the waves become patchy, and/or dissipate.
Also, they would be vulnerable to electrical interference.
I think the closest thing we might get to old fashioned telepathy is something like the tseheylu from avatar.
No words, no language, just the instantaneous transfer of data from one nervous system to the other by plug in. So you can just have your creatures holding tentacles, and signaling each other as if to say, “plug me in, I need to talk to you,” and depending on how technological your creatures are they could have an implant that can make their “plug in” telepathy into “wireless.”
One that I've not seen gone into depth here is electromagnetic fields, as distinct from "radio".
Marine animals from clams on up either inadvertently or deliberately produce electric signals, and those signals are detected through the water by marine predators (sharks, skates, rays, reedfish, sturgeon, etc). Most have their electroreceptors in ampullae of Lorenzini, and those of sharks are sensitive to 5 nV/cm.
To work at reasonable range, this requires water or conductive ground, however; air is a reasonably good electrical insulator, so I suspect such communication over air would be necessarily short-range.
We can calculate the range and data rate we'd get by comparing to an electric eel.
To emit electricity, electric eels use electroplaques which, arranged in series, emit 860V at 1A and 25Hz.
A shark can detect 1/172,000,000,000th of that voltage over 1cm. So if we can find the sphere that has 172 billion cm surface area, that's the max theoretical distance they could communicate, assuming perfect conductance. That's a 1170m radius (nearly 3/4 mile), so that'd be the max communication range with biological organs that exist on earth.
On a flat plane (eg wet ground), the distance would massively increase because now it's just a disc, not a sphere, to a shade under 274km.
Be warned that this is assuming perfect conduction, not taking into account resistance, impedance, interference, noise, and other confounding effects, all of which would severely decrease this range. akin to saying "The blue whale can emit at 188dB, and the most sensitive hearing of any animal is (I don't know, and have no idea why this is so hard to google) dB, therefore using speech we could communicate out to thousands of miles if we assume air transmits sound perfectly".
It's also assuming that it's OK to electrocute the person next to you in order to shout across to the next room.
At 25Hz, you can transmit a signal of at most 25 bits per second, which is still plenty for speech, with human speech having a bit rate of some 39 bits per second. Cranking it up to 40Hz to get the same rate as speech doesn't seem infeasible, or even higher, plus you can use other modulation schemes to get higher data rates at the cost of range.
Detection of magnetic fields (magnetoreception) has been shown in everything from bacteria on up. Many such animals have been shown to navigate by magnetic fields, for example. But for the most part, we only have hypotheses about how this is done, and don't even know the sensitivity-levels of the sensors.
The study of this area is magnetobiology, and as that page admits in its lede:
Biological effects of weak low frequency magnetic fields, less than about 0.1 millitesla (or 1 Gauss) and 100 Hz correspondingly, constitutes a physics problem. The effects look paradoxical, for the energy quantum of these electromagnetic fields is by many orders of value less than the energy scale of an elementary chemical act. On the other hand, the field intensity is not enough to cause any appreciable heating of biological tissues or irritate nerves by the induced electric currents.
That is to say, we not only don't know how animals detect this (though there are plenty of hypotheses), but we don't even have a clue how it's physically possible for them to do it as well as they do.
The page also makes the (unsupported) claim:
Their perception can be on the order of tens of nanoteslas.[citation needed]
So if you want more magic-and-woo SciFi, magnetic stuff is good. If you want Hard Science, maybe best to stick to electric stuff. But remember that they're both just two sides of the same coin! An electric field creates a magnetic one, and vice versa.
Using real world science? Not really possible like you might want it, as far as I am aware. Telepathy and telekinesis are essentially just sci-fi magic,and there's no such thing as a signal which just appears in the target's mind (unless maybe we enter the realm of quantum physics, in which things begin to get rather interesting, but still).
So let's break it down the classic telepathy you seem to want and see how your "sign language" limitation doesn't apply if you want communication:
In classic telepathy, what do we see? We see people communicating only with their minds, without requiring any kind of device or even their voices. In some cases, this communication can even be used to transfer knowledge and ideas rather than a phrase in a specific language. Do we see this anywhere in real life? Yeah, in radios, Bluetooth sharing and the internet.
Truth here is that in all these cases we see exactly that, with some differences: in radio communication you speak to the radio, the radio converts what you said into a signal and sends this signal. The other radio "magically" receives the signal, interprets it and uses it's sound box to convert the signal into something the other person can understand. Now, notice how the radios can communicate between each other without needing to use sound? That's called electromagnetic waves, a kind of wave that isn't unlike light, but it's one we can't detect with our senses alone. A very similar concept is used when you use Bluetooth to transfer a document from a device to another.
When you break it down, telepathy is no different from someone who is unaware of the existence of Morse code hearing it. The difference here would be that the beeps are in a frequency this person can't perceive.
So summing up, can telepathy like you probably mean exist? Yeah, so long as you understand it doesn't have infinite range and your species can somehow produce and detect electromagnetic waves as a means of communication (example involving the use of radio waves). If you don't want to use radio, you can just look at other alternatives in @Slows 's answer. Can information be exchanged through means other animals might not detect? Yes. Can it be Done without requiring the information to travel through some medium, just appearing at the receiving end? Yes, with magic or with sufficiently advanced technology, which are basically the same, except one category is named magic and the other sci-fi.
Others have already mentioned radio, but perhaps your creature has brain circuitry to communicate through electromagnetic induction, like a transformer. Maybe a neuron coil wraps around the amygdala and modulates the signal with their emotions, which would induce the same current in the brain of a peer. The range would be really short though, probably even have to touch heads for the clearest signal, but it also is constantly broadcasting / receiving. They'd also probably give intense spiritual significance to minerals and the planet, also being great at navigation by "listening" to the magnetic fields. If they are intelligent and develop technology, cell phones might interface with the brain this way. They'd be extremely vulnerable though without technology, get a strong enough magnetic flux and the electric current will fry their brains.
You could also make up some kind of sci-fi "quantum entanglement" explanation.
True telepathy would basically be organic radio - somehow your critters would have to be able to generate, modulate, receive, and interpret EM waves. That would allow them to communicate over distance, would be undetectable to others (without equipment scanning the particular EM band they communicate over, anyway), and depending on the band could transfer quite a bit of information.
Given that we have Earthly species that can generate and store electric charge, I don't think it's flat-out impossible for some species somewhere to evolve a way to use that charge to generate EM waves, although I can't even begin to imagine the selection pressures that would favor such a development.
One more possibility - usage of a field existing in extra dimension.
In 4th dimension (sush, string theory!) objects might be in a good proximity to each other in curved 3d spacetime so that creatures can utilize some field that leaks into this fourth dimension for their communication.
