BMR (basal metabolic rate) is the calories burned to simply keep the body alive. It doesn't include things like rising from the bed, brushing your teeth, walking downstairs, getting dressed, etc. It is basically the energy your body spends keeping you alive if you were lying down in stasis all day.
TDEE (total daily energy expenditure) at maintenance (i.e., calories burned with a sedentary lifestyle) is what incorporates generally living and going about your day. Your TDEE is adjusted with your exercise level, but it will always be higher than your BMR. That said, you should never eat below your BMR to ensure that you're eating enough calories to keep your body running efficiently.
In order to maintain weight, eat at TDEE maintenance (sedentary). To lose weight, eat below your maintenance TDEE, but above your BMR. To gain weight, eat above your maintenance TDEE.
If your maintenance TDEE is 3100 and your BMR is 2600, you'll want to eat between 2600 and 3100 calories a day to safely lose fat.
Please keep in mind that both BMR and TDEE will decrease as you lose weight, so it is important to consistently check at least your maintenance TDEE every ten pounds or so to ensure that you are continuing to eat at a caloric deficit.
As you increase your exercise level, your TDEE will increase to incorporate that energy expended through exercise (which is why you shouldn't "eat back" your exercise calories), but BMR remains the same regardless of exercise level.
