When you hear momentum, one means the linear momentum of a particle. It's a measure for how much a particle moves. Mathematically (and according to classical mechanics): $p=mv$, in words: mass times velocity.
It's intuitively a very useful concept: comparing two objects with the same velocity, but different mass, one would easily say that the one with the greater mass is "more moving", one needs more work to move the heaviest object. So linear momentum is some kind of a weighed velocity.
Another useful concept is conservation of linear momentum. Consider two objects, one that has no momentum (the objects stands still), and one that moves. Assume that when the collide, they'll stick together (this is an inelastic collision). The total momentum of the two objects together, will be the same as the original momentum of the single moving object. So now both objects will be moving. Since you want $p=mv$ to be constant (according to the law), $v$ of the two objects together will be lower, since $m$ of the two objects is larger than $m$ for only one object. I think this should be clear intuitively too.
As you stated, a photon has a momentum too, although it has no (rest) mass. The definition here is $p=h/\lambda$, with $h$ the constant of Planck, and $\lambda$ the wavelength of the photon. To explain the correlation between these two definitions, is intuitively a little more difficult, since one would need theory of relativity. But that's why I mentioned an inelastic collision: when photons collide inelastically on an object in space (which means the object absorbs the photons, so the object and photons will "stick together" after collision"), they'll give extra momentum to the object. So the object will accelerate, which proves experimentally that photons have a momentum.
To conclude, linear momentum has actually nothing to do with spin. The reason why you're probably confused, is there's also something called angular momentum. It describes how much a particle rotates (like the linear momentum describes how much a particle moves). And there exists a conservation law of angular momentum too. It's because of this law, that eventually one came up with the spin of a particle, also called spin angular momentum. It's an intrinsic characteristic of a particle; if it wouldn't exist, e.g. conservation of angular momentum wouldn't apply for the electrons in an atom. You can imagine spin of an electron as if it was rotating around it's own axis, although that's not a real accurate description.