They are mainly used to prove theoretical results since they are relatively simple (in operation) and equivalent to the modern day computer for problem solving.

Turing machines are used to prove some problems cannot be solved using a computer. One of the most famous of these kinds of problems is the Halting Problem.

Halting Problem -- Given some arbitrary Turing machine, called T, and some arbitrary input, called w, is there an algorithm to decide whether T halts when given input w?

At first reading, this may seem like gibberish, but think of T as your program, w as the data to your program (w is for word), and an algorithm as just another program (another Turing machine). This should be familiar to you; a compiler (a program) takes your program as input and performs a task. So, the Halting problem says, given a program with data, can you write another program to detect if your program normally terminates? Or, another way of thinking of it, could you detect an infinite loop by just examining the program and data?

Certainly sometimes, in some situations you could. Be nice if you always could run your program and data through T and it pops out and says, buzzzz, infinite loop. Unfortunately, there is no such T. Thus, we get a well known result:

Theorem: The Halting Problem is not solvable.

Meaning, there does not exist a TM (a program) to detect if your program will always terminate normally.

The proof of this is quite nifty. It assumes there is a TM which can solve the Halting problem, call it the "Halting Problem Answerer" (HPA). Since you can feed the HPA any program and any data, you feed it itself both as T and w, program and data. After a bit of algebraic manipulation, we encounter strange things happening, and get a contradiction.

With the Halting Problem, the contradiction ends up saying our proposed solution, the HPA, halts if it loops, but also loops if it halts. Thus we conclude no such TM can exist. (See the Turing machines notes for details.)