As you say 우리 means "us" or "we".
우린 with "ㄴ" added at the end is a contraction of 우리는 - 리+는 becomes 린.
는 (or 은 which is similar) is what they call a particle, or marker. A "topic marker" in this case.
They are necessary to make a sentence like 우리는(or 우린) 학생이다 (We are students).
But the tricky thing is that often times 는 (and other similar particles) may be omitted.
So you see both 우리, 우리는, and 우린 equally frequently.
Same goes for other similar cases, like 나/나는/난 (I am...), and 너/너는/넌 (You are...), and more.
You need to know the particles and how certain conjugated forms are shortened to get the hang of these things.