Make sure you buy one of the EyeTV products that are HD ready - you do not want one of their tuners that is analog only as the analog TV signals are now scheduled to go away in 2009.
They currently have 3 devices that fit the bill for us US customers - the Hybrid, 250 Plus and the HD Homerun. Each with it's own set of features.
For connecting an antenna to pick up your local TV stations (aka over-the-air or ota) any of the three of them will provide this functionality by connecting a coax cable to the EyeTV and then using it's built in tuner to tune in your TV stations.
You can also connect a DirecTV, DishNetwork, or cable box to any of the three of them via coax. However, via coax, you will only be able to view the unencrypted channels. And, no, I do not know which ones are unencrypted - you will have to check this out with your provider.
For viewing HD content from any of these boxes or connecting your Xbox, DVD player and other devices - you will need to go with either the Hybrid or the 250 Plus. Both of these allow for connections from both composite and s-video. None of them have component nor HDMI connections. So, this means when HD content hits your computer, it will no longer be in HD as these cables are not capable of passing through an HD signal. Stay away from composite connections, use s-video. Have been using s-video from my DirecTV box and now DishNetworks boxes for many years on my Win machines and it will give you a decent picture.
Now, as Dooley has suggested, it's time to read up on them yourself to find the one that is right for you. Check out
this page on ElGato's site.
And don't worry about compression yet. It's a little pre-mature to worry about what to do after you get the signal to your computer, as this will be the same no matter which device you end up using.