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Regular expressions are a very concise way to specify most pattern-matching and -replacement problems, and regexp engines can be very highly optimized.
If you wanted to do the same job as even a relatively simple regexp, you'd have to write a lot of code, which probably would contain a number of bugs, be hard to understand and perform badly.
Whereas doing the same with a regexp is much shorter, almost certainly performs as well as is technically possible, and is easier to understand to anyone familiar with regexpes (though it should be commented in either case)
Jon and Sqook gave a fine explanation and definition of Regular Expressions, and for simple problems it is pretty understandable, but if you use it for complex problems regular expressions can be a &$@( (at least for me ;-))
I use Expresso a lot to help me build complex regular expression code.
http://www.ultrapico.com/Expresso.htm
It has a build in library with expressions you can use, a design mode where you can build your code and a test mode where you can test and validate the code. It helped me build and understand complex expressions better!
Goodluck!
Example of critical use is JavaScript:
If you need to do search or replace on a string, the only matching you can do is a regular expression. It's in the JavaScript API on those string methods...
Personally, I mostly use regular expressions only when I need some advanced matching in some automated find/replace in a text editor (TextPad or Visual Studio). The most powerful feature in my view is the ability to match a pattern that can be inserted in the replace.