My planet orbits two suns. The orbit of the first sun is similar to Earths orbit around our sun. The orbit around the second sun however makes it appears only around the horizon. Is this possible?
Here's another option: the second sun is actually the primary sun. It's a small red dwarf, to which the planet is tidally locked--but which it nevertheless orbits at a sufficient distance to receive insufficient warmth. For a large ring of the surface of the planet, that star will only ever be visible near the horizon. That whole star+planet system then orbits around a sunlike star much further away, which would rise and set normally everywhere.
This is not possible for all locations on the planet, but might be possible at some locations. For example the Sun always appears very low on the horizon in the Arctic and Antarctic, but is sometimes directly overhead in the tropics.
The second sun would have to have an orbital inclination with respect to the plane of the ecliptic that exactly matched the planets axial tilt. In this way the second sun would appear to continuously move around the horizon as viewed from the poles, but would increasingly dip into a day and night cycle moving away from the poles and at the equator it would appear to cross the sky at a high angle.
A complicating factor is that the planet would orbit around the centre of gravity of the system as a whole (both stars) so the planets axial tilt might have to be adjusted to make this work.
There is also the issue of orbital stability. Such a system might prove to be unstable, depending on the exact masses of the planet, the stars and their respective orbital paths. Many such arrangements are unstable.
Is a planet capable of existing in a two-star system? Sure.
However, a planet cannot orbit two stars simultaneously (edit: but they can orbit a binary star system, as pointed out by @M.A.Golding). Orbits are a result of gravity pulling on an object that is attempting to accelerate perpendicular to the direction of gravity, as shown quite neatly below:
For this reason, your planet cannot 'fall' in between two stars. The star with the greater gravitational pull will keep it locked. In the event your planet reaches a point where the resultant gravitational force is zero during its orbit, it will fly off at a tangent, and the star with the greater gravitational force at this point will pull it in. This is complicated astrophysics, and likely unsustainable.
EDIT: It's also worth saying a planet caught in between two stars would be torn in half. The gravitational fields would be opposite in direction, so you can also be certain a planet would never have naturally formed here.
To answer your question though, perhaps your first sun could be the larger, and your second sun could orbit this independent of your planet (i.e. they don't orbit each other). Hypothetical systems like Nemesis show how you could make this work, and with some thinking you could definitely position your orbits in such a way that the second, smaller sun was only visible on the horizon (at certain times). Again, you need to really think it through, but it's not beyond the realms of possibility. Good luck!
There is one way this could be true for a portion of your planet: if it's orbiting in/near a leading or trailing Lagrange libration point (L4 or L5), and is tide locked to the primary star.
There are some limitations on a Lagrange orbit, however: the primary must be at least about 80 times as massive as the secondary, and the planet must have negligible mass relative to the to heavier objects (easy, even if the secondary is an M9 barely red dwarf).
Where this falls down is that in order to be tide locked to the primary, your planet would generally need to have a close, short-period orbit, but that orbit must have the same radius as that of the secondary star, and these orbits are only in the habitable zone if the primary is a dwarf star with low luminosity. If that's the case, the 80:1 mass requirement pushes your secondary down into the brown dwarf or even sub-brown (planemo?) category -- "shining" only by radiating leftover heat of formation or from low-level, unstable deuterium fusion (short-lived by cosmological standards).
The other possibility is for a planet in this kind of orbit to have a "day" close to the length of its "year" by happenstance. Venus comes to mind; it's nothing like tide locked, but has a solar day of about half of its year (combine a sidereal day longer than the year, and retrograde rotation). You could pretty readily hand-wave a planet with sidereal day very close to its year, and the "horizon star" would very slowly circle through the sky, along with the sun, in a months-long day-night cycle.