What does it mean for a browser to be "webkit-based" or "gecko-based"?
To add to the other answers, some things which differentiate the engines:
- WebKit is built to be suitable for a variety of environments, including mobile and embedded systems.
- Gecko includes an application-layer language based on web technologies called XUL which can be used for cross-platform GUI development (and is employed in Firefox, Thunderbird, and add-ons for both).
- WebKit has gained popularity and helped to push a variety of new web standards, most notably the
<canvas>drawing layer—now adopted by every major engine besides Trident (the IE engine). - Gecko probably bears the greatest responsibility for reigniting browser innovation, following a stagnation when Netscape lost the "browser wars" to IE.
- While WebKit tends to have a speed advantage, Gecko has been making great strides in this area since version 3.0, as well as minimizing resource requirements (tending to require less memory over time than WebKit, depending on implementation).
Edit: And a little history and some other browsers using the engines...
Historically, WebKit is derived from KHTML, the engine in Konqueror. Some of the changes in WebKit have been back-ported to KHTML.
Some other browsers (and applications and OSes) using WebKit include Epiphany (for *nix), OmniWeb (the first OS X browser, originally on NeXT), Adobe AIR (a runtime for using web technologies to develop "desktop" applications) and Palm's WebOS.
Gecko began development at Netscape, with the plan to eventually integrate it into the Netscape browser. The Mozilla foundation was spun off from Netscape, and proceeded to develop Gecko for the Mozilla suite, now known as SeaMonkey. Firefox (originally Phoenix, then Firebird) was built as a lightweight Gecko browser without non-browser-related functionality in the Mozilla suite, and with a defined extension API for adding features.
Some other browsers using Gecko include K-Meleon (for *nix), Flock (specialized for social networking) and Camino (Gecko in a native OS X Cocoa UI).
Evolution of Gecko.
First there was the Netscape browser.
At around the time of Internet Explorer 3, Netscape started working on a new browser engine, intending to use it for Netscape Communicator 5.
Netscape Communicator 5 never happened, but the new browser engine, now called Gecko, went on to power the Mozilla browser, as well as Netscape 6 and 7.
Gecko, now open source, is used most notably in the Firefox browser and Thunderbird mail suite, and runs on a wide range of platforms including mobile.
Evolution of Webkit.
First there was the open source Konqueror browser, powered by its own browser engine it called KHTML.
Apple used KHTML as the basis for their browsing engine, called it Webkit, and it powers their Safari browser.
Google first used Webkit for their Android browser, and then went on to use it for Google Chrome. Webkit was also used for the Nokia Browser (on Nokia S60), Epiphany, WebOS browser (on Palm Pre) and many more.
Since late 2013 Google transitioned from using Apple's Webkit to its own independent fork of it called Blink (though, Google Chrome on iOS had to continue to use Webkit due to Apple platform restrictions). Blink is/was similar enough to Webkit in behaviour that some people continue to group the two together.
Webkit continues to be developed and is still used in Apple's Safari and a wide range of open source browsers.
Other browser engines
Trident, from Microsoft, which powered Microsoft Internet Explorer versions 4 through 11, and served as the basis for EdgeHTML, which powered early versions of Microsoft Edge. Microsoft Edge is now a rebranded version of Google Chrome, complete with its Blink engine.
Presto, from Opera, which powered Opera versions 7 though 15. Opera is now powered by Webkit.
WebKit and Gecko are two different rendering engines that browsers implement to display HTML. Safari and Google Chrome use WebKit. Firefox uses Gecko.
Mozilla uses the Gecko engine, Chrome and Safari use WebKit. WebKit is supposed to have a smaller footprint.
You can find out what browser you're using (and useful tweaks) at http://www.whatbrowser.org/en/
The graphical timeline of browsers at wikipedia may be helpful: http://en.wikipedia.org/wiki/Timeline_of_web_browsers#Graphical_Timeline
As would the explanation of Gecko and WebKit
Edit: To address the edit of the original post "what do they mean", the whatbrowser.org site links to a lengthy comparison of web browsers. While this is not technically related to Gecko vs. WebKit, it is a great way to see how some of the browsers which implement the two engines are.