Where does Firefox keep cookies?

Firefox generally stores cookies here:

C:\Users\User name\AppData\Local\Mozilla\Firefox\Profiles\some random list of characters.default

An easier way to open this:
WIN+R (key combination), and then:
%APPDATA%\Mozilla\Profiles\Firefox\Profiles\randomstring.default\

(You can select the folders one by one in the run dialog. Just start typing the address.)

The cookies are in the folder mentioned by Corporate Geek (%APPDATA%\Mozilla\Profiles\Firefox\Profiles\randomstring.default), in the file cookies.sqlite.

To manage individual cookies you have to open Firefox and go to "Settings\Privacy\remove individual cookies"

A cookie is information stored on your computer by a website you visit.

In some browsers, each cookie is a small file but in Firefox, all cookies are stored in a single file, located in the Firefox profile folder.

Cookies often store your settings for a website, such as your preferred language or location. When you return to the site, Firefox sends back the cookies that belong to the site. This allows the site to present you with information customized to fit your needs.

At the top of the Firefox window, click on the Firefox button, go over to the Help menu (on Windows XP, click on the Help menu) and select Troubleshooting Information. The Troubleshooting Information tab will open. Under the Application Basics section, click on Show Folder. A window with your profile files will open.