Empty the Trash if you haven't already done so. If you use iPhoto, empty its internal Trash first:
iPhoto ▹ Empty Trash
Do the same in other applications, such as Aperture, that have an internal Trash feature. Then restart the computer. That will temporarily free up some space.
According to Apple documentation, you need at least 9 GB of available space on the startup volume (as shown in the Finder Info window) for normal operation. You also need enough space left over to allow for growth of the data. There is little or no performance advantage to having more available space than the minimum Apple recommends. Available storage space that you'll never use is wasted space.
When Time Machine backs up a portable Mac, some of the free space will be used to make local snapshots, which are backup copies of recently deleted files. The space occupied by local snapshots is reported as available by the Finder, and should be considered as such. In the Storage display of System Information, local snapshots are shown as Backups. The snapshots are automatically deleted when they expire or when free space falls below a certain level. You ordinarily don't need to, and should not, delete local snapshots yourself. If you followed bad advice to disable local snapshots by running a shell command, you may have ended up with a lot of data in the Other category. Ask for instructions in that case.
You can more effectively use a tool such as OmniDiskSweeper (ODS) or GrandPerspective (GP) to explore the volume and find out what's taking up the space. You can also delete files with it, but don't do that unless you're sure that you know what you're deleting and that all data is safely backed up. That means you have multiple backups, not just one. Note that ODS only works with OS X 10.8 or later. If you're running an older OS version, use GP.
Deleting files inside an iPhoto or Aperture library will corrupt the library. Any changes to a photo library must be made from within the application that created it. The same goes for Mail files.
Proceed further only if the problem isn't solved by the above steps.
ODS or GP can't see the whole filesystem when you run it just by double-clicking; it only sees files that you have permission to read. To see everything, you have to run it as root.
Back up all data now.
If you have more than one user account, make sure you're logged in as an administrator. The administrator account is the one that was created automatically when you first set up the computer.
Install the app you downloaded in the Applications folder as usual. Quit it if it's running.
Triple-click anywhere in the corresponding line of text below on this page to select it, then copy the selected text to the Clipboard by pressing the key combination command-C:
Launch the built-in Terminal application in any of the following ways:
☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
☞ Open LaunchPad. Click Utilities, then Terminal in the icon grid.
Paste into the Terminal window by pressing command-V. You'll be prompted for your login password, which won't be displayed when you type it. You may get a one-time warning to be careful. If you see a message that your username "is not in the sudoers file," then you're not logged in as an administrator.
The application window will open, eventually showing all files in all folders, sorted by size. It may take a few minutes for the app to finish scanning.
I don't recommend that you make a habit of doing this. Don't delete anything as root. If something needs to be deleted, make sure you know what it is and how it got there, and then delete it by other, safer, means. When in doubt, leave it alone or ask for guidance.
When you're done with the app, quit it and also quit Terminal.