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][1]. 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. 

  [1]: https://en.wikipedia.org/wiki/Long_line_(topology)