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Number theory is the study of the numbers. Pretty evident from the name. You may be asking, "What is there to study about numbers themselves?" This is a perfectly valid question as the school curriculum goes far enough to cover prime and composites, factors and multiples, but stops there. There is actually much more depth to number theory, which we will find out about throughout the next topics.
Up till several years ago, number theory was the most "unapplied" branch of mathematics. That is, it had very few uses in real life. Nowadays, number theory is important because cryptography takes advantage of the hardness of the factorization problem (e.g. it would be very very difficult to factor the product of two 100-digit primes). However, this makes us wonder "why did people study this subject, when it had hardly any uses outside of mathematics?" The answer is very simple: number theory is simply fascinating.
In order to demonstrate, I will present Fermat's little theorem (pronouned more like "ferMOT" rather than "ferMAT"). We all know that is a prime number. We're going to find the remainder when we take a number and raise it to the th power and divide it by . We get
If you have never seen this before, you should be mind-blown at the moment. This works for any prime. For example when divided by gives a remainder of .
This amazing theorem may give the impression that mathematicians have already figured out everything there is to figure out in Number theory. However, there are many things that we have not fully understood yet, e.g. The Riemann Hypothesis, L-functions, prime numbers, Birch & Swinnerton-Dyer Conjecture, etc. If someone figures this stuff out, many mathematicians would be mind-blown themselves!
In order to be mindblown further, you will have to look into the vast subject yourself. Just scroll down and press "next topic". I wish you luck in your journey.