Low brake pedal and poor brake performance could be the result of any one of the following:
Air in the brake lines;
Low brake fluid level;
A leak in the brake lines or slave cylinders;
Defective master cylinder.
The reason it is more pronounced when the engine is running is because of the power assist provided by the brake booster, but most likely you would be able to push the pedal to the floor with the engine off too if you used enough force.
Since you stated in your original post that you 'don't have much experience with this kind of thing' I strongly suggest that you seek the advice of a professional mechanic or a trusted friend who is very good with cars. A car's braking system is not something you want to take chances with.