I think we can go two ways with the question.
The first way is to assume "sense" refers to a set of inputs that we perceive and process as some type of information. So vision would be considered a single sense, smell a different sense, etc. In this case, humans have sight, touch, taste, smell, hearing and balance as base senses, along with the possibility of electric and magnetic senses most of us don't pay attention to (if they exist at all). And all the subdivisions of those senses, such as touch giving us sensations of temperature, texture, pain, pressure, stroking, etc.
This way, the question becomes too broad, as there's really no limit to things. An alien could sense magnetic fields in a way comparable to our sense of vision, while EM radiation could just be felt as some kind of environmental pressure. This alien might see a server room, with all it's tiny little magnetic fields forming and dissipating at high frequency, as a disco dance floor, while only being mildly aware of the difference between a brightly-lit field and a dark cave.
Another way is to look at what we're actually perceiving, regardless of how we perceive it. So sight is some method of perceiving EM radiation, touch is a method of perceiving direct pressure, sound is a method of perceiving mechanical pressure waves, etc.
In this case, the question is still going to be broad, as there are a lot of grey areas. Sound is ultimately just a form of touch, for example. But we could list a few basic things to detect:
- EM radiation.
- Direct pressure.
- Chemical interactions.
- Electric fields.
- Magnetic fields.
- Radiation. (Shamelessly stolen from Bryan McClure's answer.)
Of course, these things could be perceived in any number of ways, affected by the spatial, frequency, and amplitude resolutions of the sense in question. An alien who has multiple vibration sensors touching the ground could sense the phase difference between the incoming vibrations to determine the direction and distance of the vibration's origin, for example. With enough sophistication, it could make a 2D map of the ground in a wide radius by inferring reflections, frequency shifts, etc.
And the senses aren't stand-alone. Our sense of spatial awareness pulls information from both sight and sound on a regular basis, while having negligible input from our sense of taste. An alien could potentially smell the "color" of a tree as being distinct from that of a rock, and combine it with some type of echolocation to paint a picture of its environment more detailed than either sense alone could give.
And I think this is how you would make an alien seem alien. It's not that an alien is going to see using some mystical magic field we've never heard of, but that the alien senses its environment using the same senses we have, in ways we wouldn't normally think about.
On Earth, all the magnetic sensors we know of use our planet's magnetic field as a means of north/south orientation. If the alien lives on a world with rapidly changing magnetic fields, it could use the fields to track prey, determine the time of day, communicate with friends, intimidate foes, decide a cold front is coming in, etc. And the possibility of a world with a globally-aligned magnetic field used for direction sensing could elicit a response of "Whoa dude, what kind of alien world would that be?"; "I know, right? We should lay off these mind-altering light patterns."
Mystical Magic Fields
Of course, if you want to introduce magical effects, there's no real problem. It's your story, after all. Just try to keep them consistent. And a good way to do that is to start with human senses, and figure out how they work.
You want a magic field that tells your aliens when other aliens are angry? Well, make it so certain chemicals that cause the "anger" emotion release magic "photons" of certain frequencies. Of course, it's probably more complex than that, since there's unlikely to be one, single catalyst for anger. Instead, the aliens would see (feel/smell/whatever) a combination of magic photons of different intensities and wavelengths. Just like "sky blue" is a fairly even mix of green and blue with about half as much red but humans just see a color, "anger" could be some particular mix of adrenaline, various endorphins, and so forth.