it being the only of the 4 fundamental forces not fully understood.

Actually, no. Gravity is fully understood, explained as it was in 1905 in general relativity. What we cannot do, however, is unify this theory with quantum mechanics, the other theory that has dominated science. That is where all our methods fail, as it leads to mathematical infinities that crop up.

One theory says that space is curved.

Just nitpicking here. It should be space and time, according to general relativity. Both these quantities curve together to form curved space-time, forcing objects travelling in a straight line to adopt a bent path.

This theory seems to imply that space has a texture.

Not necessarily. By space-time curvature, this does not mean literally curved; it is simply a good analogy for the mathematical version of what happens. Space and time, in the mathematics, are said to be curvbed; this does not apply physically.

Space and time are said to curve because the mathematics treats them as such. In reality, however, you would be hard-pressed to say if space-time were actually curved just by looking.

I can understand this 2 dimentionally, but how would this work in 3 dimensions?

Think of a sphere. You have just imagined curved space-time in three dimensions.

Earth is resting on the edge of "space".

No. Think of Earth being inside a sphere. It will then have to follow the path of the sphere i.e. the curve of the sphere. Earth is in space and time; its motion is simply being affected because it measures space-time in such a way that it appears curved and follows a curved path.

How do moons orbit planets then? Shouldnt the moons be orbiting the sun also? Am i missing something? Does three dimentional space allow space to warp in a different way from what im thinking?

You forget that planets also curve space-time. In fact, anything with energy curves space-time. As a result, the moon is now following the curvature of space-time caused by the Earth, forcing it to orbit the Earth.

And the alternate theory, gravitons.

Quite correct. However, a collection of gravitons can be shown to be mathematically equivalent to curved space-time; see Wikipedia's article on it.

If gravity is always attractive, how would gravitons work? (This kind of relates to the other force particles as well)

That is why gravitons are troublesome creatures. They attract everything, even other gravitons. When it does so, however, we have an infinity of gravitons appearing, because the original gravitons exchange gracvitons, which in turn exchange gravitons, and so on.

I understand repulsion: a matter particle emits a force particle and recoils back, another matter particle aborbs the force particle and changes its velocity in the other direction.

Well, the answer to your qualms about attraction are hidden here. The second matter particle simply changes its velocity in the direction of the first matter particle. Simultaneously, it also emits a particler to the first matter particle, forcing it to the same thing it has done. As a result, both of them attract each other.

If the first theory is right, then would gravity still count as a fundemental force? Since it doesn't require force particles at all, shouldn't it be classified as something else?

Gravity is always a fundamental force. Don't worry.

And lastly, is there any way that both theories are right?

General relativity was long demonstrated to be correct. Unifying it qwith QM is tricky, thjough,.