In many situations you can use regular expression.. I am assuming that you already understand the meaning of regular expression and its usage.. basically you can use them anywhere, where instead of validating a static string you need to check some string which will be generated dynamically and also follow some pattern.. for example you have to validate some greeting message application give after login.. if your application gives message as 'welcome visitor' you can validate it as a whole, but if your application gives message as 'welcome YOURNAME' , YOURNAME might vary based on the credential etc. In this case you can just validate that welcome is followed by some characters instead of static string.. this is very simple explanation but if you understand the concept you will find its usage at many places..
As a tester unleass you look to write some code in the form of scripts to perform any kind of automation you will mainly not venture into this area of regular exp.
But there may be cases where you may use wild characters while testing your application which may come under this category.
Suppose I want to retrieve all the records for account name starting with A (instead of searching individually), then if your app permits wild char search then we may use something like A* to retrieve all matching accounts.
Let me give you a simple example, you have a file.txt which consists of some some 100 email ids and you want them to be one after the other in a column so now you go to replace and mark the regular expression then in FIND WHAT type , replace with "\n".
This is small example where we can save a lot of time with regular expressions
please read the post of jonty, well another example is that, when we record on a web site,we get many requests for suppose you have to remove the unwanted requests(.gif,.png, etc). we can use regular explessions for saving the time.