Theoretical probabilities are your calculation of what "should" happen, when you are able to describe all the equally likely outcomes. Then the theoretical probability is the number of "successes" (outcomes that fit the probability you're trying to find) divided by the number of equally likely outcomes. For example, if you roll a fair six-sided die, there are six equally likely outcomes: 1, 2, 3, 4, 5, and 6. The probability that you roll an odd number is 3/6 (equals 1/2) because three of those outcomes - 1, 3, and 5 - are odd, which makes them "successes."
Empirical probabilities are your calculation of what "did" happen in an experiment. Again, you put the number of successes in the numerator, and the denominator is the number of trials you did. If I roll a fair six-sided die ten times, one possible result is that I roll 4, 2, 5, 6, 3, 2, 3, 5, 2, 2. My empirical probability of rolling an odd number in this case is 4/10 (equals 2/5).
Notice that the empirical probability is not necessarily equal to the theoretical probability. Notice also that if you do the experiment again, you may well get a different (but still correct) empirical probability. The theoretical probability, on the other hand, has only one correct answer.
As for professions, consider industries like insurance or casinos and name some of the people who work for those companies. Probability theory is also the foundation of statistical inference, used by statisticians and researchers of all kinds.