I want to try to buy one that's not too expensive, since money is always limited for my family.

A capture card allows you to capture the video and audio output coming out of a source - in our case a video game console. The capture goes to your computer as a video source. You can take the capture and put it in broadcasting software like OBS or XSplit where you can build a layout and then stream to Twitch, YouTube, Beam, etc. If you buy some extra hardware called a splitter you can split the input source (the console) to multiple outputs so you can play the game on your TV while simultaneously capturing it with the capture card.

How expensive one is depends on what kind of consoles you’re trying to capture.

The best standard def (composite and s-video) capture card is gv-usb2 which is relatively cheap at only $30-$40 USD. This is good for older consoles like the PS3 era and older. If you’re trying to capture HD consoles over HDMI then unfortunately you’re looking at $100-$150 where the cheaper you go the worse quality you’ll get with more delay in the capture video/audio.

If you’re using a PS4 or an Xbox One the you can use their Twitch app to stream directly to Twitch without the need to get a capture card.

As Timmiluvs said it all depends on what console you want to capture, but if it's standard definition I'd also recommend the GV USB-2. Alot of people use it and swear by it, I've literally never heard one bad thing about it :)