OpenLayers is an open source, client side JavaScript library for making interactive web maps, viewable in nearly any web browser. Since it is a client side library, it requires no special server side software or settings—you can use it without even downloading anything! Originally developed by Metacarta, as a response, in part, to Google Maps, it has grown into a mature, popular framework with many passionate developers and a very helpful community.
OpenLayers makes creating powerful web-mapping applications easy and fun. It is very powerful but also easy to use—you don't even need to be a programmer to make a great map with it. It's open source, free, and has a strong community behind it. So if you want to dig into the internal code, or even improve it, you're encouraged to do so. Cross browser compatibility is handled for you—it even works in IE6.
OpenLayers is not tied to any proprietary technology or company, so you don't have to worry so much about your application breaking (unless you break it). At the time of writing, support for modern mobile and touch devices is in the works (with many proof of concept examples), and should be in the official library in the near future—if they aren't by the time you're reading this.
OpenLayers allows you to build entire mapping applications from the ground up, with the ability to customize every aspect of your map—layers, controls, events, etc. You can use a multitude of different map server backends together, including a powerful vector layer. It makes creating map 'mashups' extremely easy.