An integer is a whole number. 5, 37, -4, 132 are all integers. 1/2 is not.
Angular momentum has is one of the quantities that (on the atomic level) comes in chunks. That is, the values go up by definite amounts, but nothing in between.
Electric charge also comes in chunks. The energy levels in an atom go by chunks.
The smallest nonzero amount of angular momentum a system can have is related to a value called
Plank's constant=
h=6.62*10^-23 J s
http://en.wikipedia.org/wiki/Plank%27s_constant.
smallest angular momentum=(1/2) h/(2 π).
Now, if the angular momentum goes like
n*h/(2 π)
where n=...,-3,-2,-1,0,1,2,3...
then we say we have an "integer" amount of angular momentum.
If it goes like
(n/2)*h/(2 π)
where n=...,-3,-1,1,3...
then we say we have half-integer angular momentum.
There is a certain quantity associated with angular momentum called spin. The cartoon of spin is the idea that the fundamental particle (like the electron) is somehow spinning about its axis. This picture isn't terrible but it really isn't true.
Anyway, some particles, like the electron, have 1/2 integer spin. They are called Fermions, after the Italian physicist Enrico Fermi.
http://en.wikipedia.org/wiki/Enrico_Fermi
Some particles, like the photon, have integer spin. Those are called bosons, after the Indian Satyendra physicist Bose
http://en.wikipedia.org/wiki/Satyendra_Nath_Bose
Fermions exclude each other from occupying, roughly speaking, the same space. Technically speaking, they can't have the same set of quantum numbers as another fermion.
Bosons, on the other hand, prefer to be social. If one photon passes by an atom ready to emit another photon of the same energy, sometimes it will do so in a way that exactly matches the first photon. If there are a bunch of photons all together, then the odds go up for an excited atom to emit a photon that matches. This is the basic idea behind the laser.
The concept of spin has subtleties that are pretty weird, one of which is a rotation of 360° only gets an electron wave 1/2 of the way around. It needs another 360° to be back in phase.
Whatever this spin is, along with ordinary angular momentum, the total angular momentum is conserved.