OpenLayers makes creating powerful web mapping applications easy and fun. It is very simple 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. A big advantage of OpenLayers is that you can integrate it into any closed or open source application because it is released under the BSD 2-Clause license. 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 by the Google Closure Library— but you need to have IE9+ because VML rendering (specific to IE8 and older versions) is no longer supported. Furthermore, OpenLayers 3.0 supports modern mobile, touch devices making it easy to develop for mobile technology.
OpenLayers is not tied to any proprietary technology or company, so you don't have to worry much about your application breaking with third party code (unless you break it). Although it's open source, you will get good support from the community and there are commercial support options as well. The library is recognized as well —established by the OSGeo (Open Source Geospatial Foundation), having passed through the OSGeo Incubation process, a kind of open source quality mark for geospatial projects.
Note
You can read further about OSGeo Incubation at http://wiki.osgeo.org/wiki/Incubation_Committee
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, and so on. You can use a multitude of different map server backends together, including a powerful vector layer. It makes creating map mashups extremely easy.