If you know anything about Internet media extenders, then you know about the Apple TV. It's biggest virtue over the other selections is that you can easily stream videos to it from your home media library. Now, if only iTunes wasn't such a miserable excuse for a media program, whether you run on on Macs or Windows, I'd be perfectly happy with this pairing.
If you want the widest possible choice of Internet video "channels," then you want a Roku box. It also features the best built-in Internet video search funciton. While the top of the line Roku 3 is quite good, you can get a lower-priced model and not give up any real functionality.
If you want to see anything you can see on a Chrome Web browser on your smartphone, tablet, or PC on your HDTV, then Google's inexpensive Chromecast is what you want.
If, like me, you also watch a lot of DVDs, you may want to consider getting an Internet-video enabled DVD player. This way you only need one box for most of your Internet video needs. The downside is that the user interfaces tend to be bad and your selection of Internet "channels" are very limited.