The Xbox 360 can indeed play games on CRTs. CRTs have had various iterations over time, though, so not all CRTs are optimal for this. For best results, I'd try to get a TV with 720p (HD) resolution support or better, plus component or HDMI in (and honestly, an LCD if you can swing it...)
That said, you can technically play games with a SD TV that has only composite in (Yellow/Red/White).There are also many component[rgb] cables available for mid tier tv's.
Do note that many games are optimized for higher resolutions, so UI elements and text might be eye-hurtingly small on a small SDTV.
If possible try to find a tv that has a 16:9 instead of 4:3 as this will reduce distortion.
Almost all Xbox 360's have HDMI out, so if yours supports this, any LCD monitor capable of 720p or better (basically, most anything you could buy in the TV or monitor section of any major retailer) is going to make a decent Xbox 360 display.Many dlp tvs can also play hdmi in high resolutions and are lighter than crt tvs for their respective sizes,but need a darker room. mini projectors may also be an option worth concidering as well.