What is a Turing machine and why do people keep mentioning it? My IBM PC is all I need to do my computation! Why does anyone care about these machines?
The amazing thing about a Turing Machine is that it is the most primary computing machine that exists and it can perform computations as powerful as any computer that exists today. There is no computing problem that you can do with your computer that you can't do with a Turing Machine. In fact, your computer is a TM if you want to look at it that way.
Alan Turing invented the TM and used a certain version of it to decode the Nazi encryption machine Enigma.
So to conclude it is a very powerful machine and has many applications in Theoretical Computer Science. (Also a super interesting and stimulating subject btw).