Clearing Private Data 
It is possible to select individual history items to delete, by using the history side bar. You can open the history side bar by selecting from the menu the View -> Sidebar -> History option. Deleting singular items can be achieved by finding the item, right-clicking on it to get the context menu, and selecting Delete. Deleting multiple entries can be achieved by right-clicking the list to focus it (for left-clicking will open the web page under the mouse, and losing focus on the list) and using the up/down arrows, and shift+up/down to select blocks of entries. Pressing Delete on the keyboard will delete the selected history items.
It is possible to delete all history items by opening the preferences dialog on the menu at Edit -> Preferences, and clicking on the Privacy tab and then the History tab. A button will be there to Clear Browsing History Now. This will clear all history items.
Under this tab also, by changing the option Remember my browsing history for X days, to 0, history items will not be kept between sessions.
Auto-completed URLs will disappear as corresponding history items are deleted.
Cookies can be cleared by selecting the Privacy tab and then the Cookies tab under the user preferences from the menu at Edit -> Preferences. All cookies can be deleted by clicking the Clear Cookies Now button. Individual cookies can be cleared by clicking the View Cookies button to open the cookie viewing dialog. Selecting cookies and pressing Delete on the keyboard will delete those cookies.
By changing the option to Keep Cookies to Until I close Firefox, cookies will not be kept between sessions. Also it is possible to choose to block all websites from putting cookies on your computer unless you have explicitly allowed them to do so, by adding them to a list accessible from this tab.
Other Data 
It is possible to delete other private data by using options under the Privacy tab of the user preferences. See Preferences for more details.
If you just want to delete one or a few values from the saved (remembered) list of field (or form) values, navigate to the item you want to clear (don't click on it, just "point" to it), then use the Shift-Delete key combination to clear it.
Deleting All Private Data 
You can delete all private data, including browsing history, cookies, cache, saved form data, passwords, and download history, by opening the Clear Private Data dialog under the menu in Tools -> Clear Private Data (or by pressing Ctrl-Shift-Delete on your keyboard). This will ask you to list your preferences as to what should be cleared from Firefox. If you want to do the same thing often in the future, you can deselect Ask me before clearing private data, and the dialog will not be cleared.
Your preferences for this option can be set in the user preferences dialog under the Privacy tab; there is a button down the bottom right that says Settings, which will effect this function.
Security Extensions 
This Firefox extension allows you to surf the web without leaving a trace on your local machine. It temporarily disables the following while it is active:
- Browsing History (Websites you've visited, no auto-complete of URL's - Cookies (Doesn't allow websites to store cookies *Warning* this could cause difficulties when trying to access user account or E-Mail) - Downloaded Files History (All the files you've downloaded) - Disk Cache (Any files normally stored on your local machine "i.e images" will not be stored for fast viewing) - Saved Form Information (Information that would be used during the save form feature in Firefox "i.e save user name and password on this computer?" will not work) - Sending of ReferrerHeader. (When you click on a link from one web page or website to another, the browser usually sends a Referer/header to the server to tell sites where you came from.)
"Can you recognize good and reliable Internet content? Did you have a bad experience that you would like to share with others? WOT can help you. Seeing website reputations on your browser allows you to learn from other people. This shared knowledge makes it easier to avoid online fraud, including phishing and spyware. WOT can add reputations also to web search results, Digg, Gmail, Wikipedia, and other selected sites." -Against Intuition, Inc.
"This of course preserves your privacy to great extent" - Anonymous
"This extension will allow you to easily control cookie permissions. It will appear on your statusbar. Just click on the icon to allow, block, or temporarily allow the site to set cookies. You can also view or clear the cookies and exceptions by right clicking on the statusbar icon. For safer browsing you may choose to deny cookies globally and then enable them on a per site basis." -Ron Beckman