Well,
Usually, there're three kinds of tests. Unit Tests, System tests and QA tests. Unit tests, as the name suggests, test small units - separate functions and classes.
For all modern development environments there're unit test frameworks. There's Nunit for .net, as well as MS unit test framework in Visual Studio, CPPUnit for C++, JUnit, etc. All meant for one thing: connect to parts of your program, run your pre-defined scripts and report error.
CPPUnit, for example, is based on macros like CPPUNIT_ASSERT_EQUAL, meant to be used as something like this: CPPUNIT_ASSERT_EQUAL(sum(arr), 17). And it would say if it's not equal, in which case the test will be considered failed.
You are supposed to develop the tests for every function, and after that - you are not afraid to change and optimize the code. It is generally referred to a "repeatability" - ability to do a complex action, such as full testing of all codebase, with a single click.
Unit tests are required for every moden software development, because the experience shows that they improve development speed. It is also suggested that unit test code may serve as a kind of "documentation" for the library.
System tests are automated tests of larger, high-level functionality. The idea of system tests is to feed clean input (such as databases, user input, etc) to test the whole thing, validating the output against pre-defined resutls. It is essential that the system ouput is deterministic, and depends only on the input. Every time the system changes, the system tests change also.
TDD is a cool-sounding bad idea, suggesting that you should not develop anything before implementing the proper automated tests, and then writing code to satisfy the tests. It is regardedd as failure, because changes in the design are inevitable during development, and a small design change usually causes drastic changes in the unit tests.
Manual QA is the final, and most important type of software testing. The idea is to prepare a test plan, whcih is done during design and coding phases, collecting all ideas developers had during coding of every if statement, how to actually make this particular if statement run along the less expected code path. The QA personnel, meant to be capable of anything that can be done with the program without development environment, can follow the resulting test procedure and their own ideas, to find more bugs.