There are seven ways to get TV these days, and your 250 can receive three of them:
Analog TV (NTSC) with an antenna the way it's been for many years. The 250 can receive that.
Digital TV (ATSC) with the same antenna used for Analog TV. The 250 can receive that.
Analog cable TV as it's been for many years. The 250 (and late model TV sets) can receive the unscrambled channels (mostly local stations), but you need a cable box from the provider for the scrambled ones.
Digital cable TV. It also has unscrambled channels (known as "Clear" QAM and again mostly locals) that can be received by the EyeTV "HomeRun" but not by the EyeTV 250. It also has scrambled channels that need the cable box AND a particular Content Authorization System access card for that provider. In your case, you need a cable box to receive both unscrambled and scrambled.
Satellite TV. You need an antenna and satellite box to watch that.
This is the breakout cable that should've come with your EyeTV 250 that you can use to connect to a cable or satellite box:
The block with two connectors are the stereo audio left and right connectors. The block with the yellow RCA female connector can connect to the Composite Video output of a cable box. The block with the multi-pin S-Video connector can connect to a similar connector on a cable box, if it has it, in lieu of the Composite Video connection, for a slightly better picture.