What are the "sleepimage" and "swapfile" documents? I was using Disk Inventory X and found a file called sleepimage in /private/var/vm/ It takes up 4 GB! I also found numerous other files in the same folder called swapfile0 swapfile1, etc. What are they and are they safe to delete so I can free up space? (almost 10 GB)
swapfile0, swapfile1 and so on are swap files. You can sort of think of them like "backup" RAM. When your computer doesn't have enough RAM to accomodate the memory requests of all your programs (which is almost always, this is normal), it uses these files to "write out" part of what it had in RAM.

The sleepimage has a very similar use, except it's used only when your Mac hibernates/wakes up.

Deleting any of the swapfile* files is likely to mess up your system, but you can safely delete sleepimage. However, it's kind of pointless, it will probably be recreated the next time you put your computer to hibernation. If you really want to delete it, you'd have to use sudo rm, since root owns the file.

Finally, it's normal for the sleepimage to take up to 4 GB; it will usually take the same space as your RAM.

EDIT You can delete /var/vm/sleepimage safely, but it will be recreated when you put the Mac to sleep. If you run sudo pmset -a hibernatemode 0; sudo rm /var/vm/sleepimage, the Mac will use a normal sleep mode (like desktop Macs by default) and it won't recreate /var/vm/sleepimage. See this post with more information.