The Hue Bridge is the brain that wirelessly controls your Hue lights. The bridge plugs into your router via ethernet and connects with your phone and light switch controllers to control your Hue lights. The Bridge is the only required dedicated piece of hardware other than the Hue light bulbs.
There are two versions of the Hue Bridge, the 1st generation and the 2nd generation. The 1st gen bridge works the same as the 2nd gen with the exception of Apple HomeKit. If you have or plan on integrating Hue with HomeKit, you will need the 2nd generation Hue Bridge. Read more about choosing a bridge in the 3 Steps to Hue Guide.
All lights and bridges are compatible. For example, 1st gen lights will work with a 2nd gen bridge. Likewise, 2nd gen lights will work with a 1st gen bridge.