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.

우리 - we, us, our 우린 -> 우리는 = 우리+는 - we