Basal Metabolic Rate (BMR) Is a measure of energy expenditure per time unit. Commonly reported in kCals/Day, BMR is the approximate measure of how much energy is required to sustain regular bodily function. (i.e. breathing,circulation, etc). BMR is often used interchangeably with RMR (Resting Metabolic Rate). Wikipedia Page on BMR
Thermic Effect of Feeding (TEF) accounts for ~10% of TDEE, and represents the increase in energy expenditure required to digest food. TEF is generally excluded from estimated TDEE because it is difficult to measure outside of a controlled setting. The abstract of this paper by E T Poehlman gives a good overview of TEF and TDEE.
TDEE stands for Total Daily Energy Expenditure, and is commonly approximated by multiplying BMR by an activity factor. This article has a complete breakdown of the components of TDEE.
If you are trying to lose weight a good place to start is your BMR/RMR. Adjust your calorie intake slowly over time based on how you feel and your average weekly weight. If you are eating below your TDEE on average, you will lose weight. Eating below 1200kCals for an extended period can be dangerous. Always consult a medical professional before changing your diet. Here is the CDC overview on healthy weight loss
Activity factors are a way of estimating TDEE from BMR. Most people with an office job would be considered lightly active or sedentary. It is really easy to over-estimate the amount of excercise you do, when it doubt start with sedentary and adjust your food intake slowly as needed. It is important to remember that these factors are not precise, and should be used only as an approximation. Example activity factors for the Mifflin-St Jeor equation.
The The Mifflin-St Jeor equation is the most accurate way of determining BMR (also called Resting Metabolic Rate) without measuring body fat percentage. Even though it is mostly accurate there is typically a +/-10% variance from true BMR. Here is analysis of the accuracy of the Mifflin-St Jeor equation.