What does i3, i7 processor mean when buying desktops?

The Intel Core i3 processor is enough for the tasks you want it to do. If hooking up computer to TV, you are best off getting an HDTV (modern these days and pretty much only choice anyway) and getting an HDMI cable and hooking it up that way. If you are gonna use the computer for high-end games like "Crysis", a good graphics card would be a better bet than integrated graphics.