There are hundreds. Some are extinct. Some are/were only used on one type of hardware. Some are only used for specific task -- like Artificial Intelligence (LISP, Prolog), economical programs (COBOL), scientific programs (FORTRAN), server-side web-scripts (PHP, ASP, JSP), client-side web-scripts (JavaScrip). Others are general-purpose (C, C++, C#, Java). Some are teaching-languages (e.g. Logo and Pascal). Then there are RAD -- Rapid Application Development -- for programming GUI-programs (e.g. Windows), like VisualBasic and Delphi.
COBOL and FORTRAN are two of the oldest languages. FORTRAN was used for scientific programs, while COBOL was used for economical programs -- like systems for banks. Although practically extinct, there are still computer-systems in banks based on COBOL (Common Buisness Oriented Language). Among it's inventors, were Rear Admiral Grace Hopper from the US Navy.
C was created parallell with the Unix operating system, and most of Unix (and Linux) OS -- as well as most of the applications that runs on it -- is written in C. C is the most "low level" of the high-level programming languages (it's "close" to the hardware), and it's possible to do thing i C you otherwise had to use machinecode (assembly language) to acomplish. It's also possible to create very fast code in C.
C++ is an extension of the C language, that adds functionality for object-oriented programming. It also adds several class-libraries (object hierachies), most importantly for input/output (keyboard, screen and files). It also simplifies the use of dynamic memory.
Qt is a class-library/framework, which make it's easy to program GUI-programs (e.g. Windows) in C++, and easely port them between various platforms (e.g. Windows, Mac, Linux).
Java is heavely influenced by C++, but is a more "pure" object-oriented language. Unlike C++, it has a buildt-in single class-hierachy, though the primitive types (integer, float,...) exists outside this hierachy. Java is geared towards security and intended to be used over web, so it lacks pointers as they introduce security-problems. It uses a combination of compiling and interperting. A program is first compiled into byte-codes, and this byte-codes is then interperted on a "virtual machine".
C# is from MicroSoft and based upon C++. It's geared towards making Windows-programs in the .NET framework. It's a good choice if you *only* intend to program for Windows.
Java and C++ is probably the two languages you will get most back from learning. I think C++ is the better language, but Java has somewhat of a hype. In any case, C, C++, C# and Java are rather similar, so if you know one, you'll easely learn the others.
PHP is used to make server-side scripts for web-pages. Programs are embedded in the HTML documents, and the web-server runs these scripts and replaces them with the result of the programs. Often PHP is linked to a database, containing information about the web-sites users. This way, the page the user in the end is sent by the web-server, can be customized and personalized.
If you intend to work as a web-designer, then learning PHP and JavaScript (along with HTML and CSS) -- and perhaps ASP, will be useful.
For system-administrators, languages like Perl, Python and Tcl/Tk, can help you automate much work. These are also good languages to write small scripts for doing useful things; like downloading email from a Yahoo-account, downloading YouTube-videos or downloading pages from some website. These are interperted languages, ideal for development through trial-and-error. Ruby is a similar language, but object-oriented. Ruby-on-Rails uses Ruby for making dynamic web-content.
If you intend to learn Object-Oriented Programming (which you should), I would like to plug SmallTalk. It's perhaps not in so much use, but it really shows you the beauty of Object Orienting, which you can take with you to other "lesser" OO-languages like C++. In SmallTalk *everything* belongs to *one single* class hierachy, even all the elements of the language itself.
Pascal is in little use, but it's ideal for learning programming, as it forces you to develop good work-habits... something you can take with you as you depart for other languages. It's also the language used in Delphi, a RAD.