Some people, like me, do not have external hard drives big enough to use for full backups using Time Machine. Although I recommend buying a big external drive, you can still use Time Machine for your more precious files on smaller drives.
First of all, launch the “System Preferences” application. Under the “System” header, select the “Time Machine” pane. Click “Options….” This will bring up a drop-down menu where you list every folder or file you do not want Time Machine to backup. Hit the “+” button and add whichever things you don’t need backed up, until there is enough room on the external drive, and hit “Done”.

Now, Time Machine will set a two minute countdown for trying to backup, and will warn you if it does not have enough room on the external drive.