This is going to be a bit long -- but very detailed, so please read it carefully. There are several reasons that an xbox 360 system will have problem reading discs. The error messages also vary from unreadable disk, no message at all but the game skips, "please insert disc in a 360 console" to "open tray." There are a few reasons this may happen:
1. Bad disc. Check the back of your disc for scratches. If it looks good (or plays fine in another console), then the disc is probably fine. Also-- try more than 1 game to be sure.
2. Bad laser. The XBox 360 has a DVD drive very similar to a PC's DVD drive inside. The drive contains a bunch of parts including a few motors and gears and a laser. The laser is the part that actually reads the disc. The reality is... with all the hours and hours of gameplay, lasers do eventually wear out. The laser isn't very easy to replace, but it is replaceable. You can find any and all XBox 360 lasers at the following website:
3. Bad DVD Drive. Sometimes other components inside of the DVD drive can become bad such as the motor that spins the disk, the motors that open the tray, etc, etc. You can get a new DVD drive to just replace everything, including the laser, in one shot but there is a catch. The XBOX 360's DVD drive is tied to the motherboard using a software lock. The easiest way to overcome this lock is to buy the exact same manufacture DVD drive that is in your system now, remove the circuit board from the old DVD drive (which contains the software key) and transfer the circuit board into the new DVD drive that you purchase. If you do this simple procedure, there is no flashing involved. The following website carries DVD drives and also provides instructions on swapping the circuit board from the old drive into the new one.
XBox 360 parts: http://store.richspsxparts.com/xb360ac.html
DVD Drive Swap Info: http://store.richspsxparts.com/imnooninxb36.html