When I'm searching for the right tv which specification should I look for that will tell me i can use it with Xbox 360. like the size, or the 1080i (whater that means)
Anything HDTV will work. Just stay away from EDTV plasma screens, they are too low of resolution to be considered high-definition.
720p has 720 lines of horizontal resolution refreshed 60 times per second.
1080i has 1080 lines of horizontal resolution. Every other line is refreshed 30 times per second.
1080p is not an "official" HD standard, but is being support by more displays. Only HDDVD and Blueray players can display 1080P currently. The 360 and PS3 will support it in the future. I don't know of any plans to broadcast 1080p picture quality. 1080p is 1080 lines refreshed every 60 seconds.
Most experts agree that 720p offers the best broadcast viewing experience. Mostly LCDs offer 1080p support, but some plasma screens are too. At anything under 37" inches you really can't tell the difference between any of the resolutions, so it's not much if an issue.
There really isn't a lot of difference between LCD and plasma screens anymore. I'm sure you'll be very happy with whichever you choose. I ended up choosing a plasma screen myself. The problems with short life and burn-in were resolved long ago. A plasma will lose about half it's brightness in 60,000 hours. Most CRT TV's lose half their brightness after 25,000 hours. How long is 60,000 hours? If you watched TV 8 hours a day, every day for 20 years, it would be about 60,000 hours. It's not recommended that you leave static images on a plasma for an extended period (just like CRTs), however any "burn-in" is usually gone after a few hours of viewing regular content. If the display will be used "primarily" for gaming, then an LCD is probably your best choice.
I chose a plasma because the color saturation and contrast were better than comparable LCDs. I also wanted a 50" screen, and LCDs were too expensive in that size range. I also didn't like the viewing angles of LCD screen. Even though they can claim 170 degree viewing, the colors tend to shift and brightness is reduced when viewing off center. A plasma will provide the same experience at any angle.