There could multiple things wrong, and hopefully nothing serious. Here is a complete list of some problems and solutions that can be wrong with the analog stick.
1. The Left analog stick may be sticky. If you eat greasy foods while playing your Xbox 360 console, try to avoid this from now on. Grease from food particles may slip into the analog stick, and make it sticky, causing the analog to move constantly to one fixed position. Please look at the next problem and solution ahead before attempting to remove your controller with a screwdriver. Here is a complete guide on how to access the inside of an Xbox controller.
2. There may be a simple glitch in configuration. Try turning off your Xbox 360 controller by holding the Xbox Guide button for approx 4 seconds untill a menu comes up. Select "Turn off controller". After doing so, try turning on your controller again. If the problem persists, there may be a greasy analog stick, or an object that broke that holds the analog stick together. Go back to problem/soltution #1.