Make sure you do your research before buying a new optical drive for your 360. You can find plenty of guides online. You must get the same model drive as the one you currently have. Once you get the replacement drive you'll need to either: extract the encryption key that is on the dead/dying drive and load it into the replacement, or replace the controller inside the new drive with the one from the old one (if this isn't the bad component of the drive).

I went through this earlier this year. The motor that spins the disc was dying and discs would not load (this meant I could not extract the encryption key, you must be able to load a game to get the key). It turns out that the model of optical drive in my Xbox had all of the wires soldered on instead of ribbon cables. I had to desolder 10 wires from both drives, remove the boards from both drives, replace the board in the new drive with the old drive's board, and solder all 10 wires back on. I recommend spending the money to have it repaired if you don't have experience working inside electronics.

Edit: To clarify, the wires I'm talking about are inside the optical drive, not the SATA and power connectors that go from the drive to the motherboard and PSU.