It is easy to see that every connected $1$-dimensional second-countable manifold (that is, what is often called just a manifold) is either homeomorphic to $\mathbb{R}$ or to $S^1$. Now let's drop the secound-countable-condition.

How do you prove that every connected $1$-dimensional manifold homeomorphic to $\mathbb{R}, S^1$, the long line or the long ray? And why are the long line and the long ray not homeomorphic?

A good survey about the latter spaces can be found in the wikipedia entry. Basically, a long ray is built up of $\omega_1$-many intervals pasted together, and the the long line consists of two long rays in both directions.