The classic yellow 'check engine' light will be on when there is a problem with the engine or emission control system. In most cases, this will not affect engine operation.
You shouldn't experience any lack of power and changes driveability. Have in mind that models for the US market usually have 'Service engine soon' light instead. If there are no other symptoms, you can safely use your BMW until you have a chance to fix the problem.
Half Check Engine Light
Half Yellow 1/2 'check engine' light indicates an engine-related problem that will affect the power output. It comes with 'Engine fault! Reduced power' message on your iDrive display.
It can be caused by the failure of various sensors or solenoids.
The engine ECU will cut down the power output to protect the engine. You might also have some driveability issues. Yet, even with this fault, your BMW will still get you home. Drive moderately and don't push the engine. Avoid unnecessary driving until you have fixed the problem.
Red Check Engine Light
Red 'check engine' light indicated a serious issue. It means that something is seriously wrong with your engine.
Stop the car as soon as it is safe to do so. Turn off the engine, let it cool down, and inspect it for unusual sounds and leaks.
Monitor the engine temperature. Don't allow the engine to overheat. Do not drive the car if the engine overheats. This can easily destroy the engine.