Swap space is a part of the disk that's used to store additional content from the memory. Since it's stored on disk, it isn't erased when the power goes out (unlike the RAM content). However, when the system boots, it doesn't read back anything that was stored in swap — the data there is not directly usable since it belongs to programs that are no longer running.
The kernel uses the swap space to store some of the data that would normally be stored in RAM. The data can't be used directly while it's on disk, so it needs to be loaded back into RAM when it is used. The point of swap is to allow the system to use more memory than is available in RAM alone. The kernel starts writing out parts of the RAM content to swap when RAM usage gets high.
Data that is in the swap space remains until it's overwritten by new data. Mostly that information is unusable because it's too fragmented and partial. However, if confidential information is written to swap, it may be a security risk — all it takes is for one key to be written there. So if there is a risk that your hard disk is stolen, you should encrypt your swap space.
Linux can use the swap space to hibernate the system. In this case, it swaps out all processes (and not just some pages as needed, as is normally the case) and writes additional data to the swap space that allow the kernel to reconstruct the full memory layout on the next boot. In addition, the information is stored in a different format (in most configurations, the hibernation image is compressed).
Swap is completely transparent to processes. A process doesn't know whether a particular location in memory happens to have been swapped out. At the time a process is accessing a memory location, that location is always in RAM. You can read from the swap partition directly, but you can't do anything useful that way.