Regular expressions allow you to formulate patterns to search for. Here's an example: It is easy to search for the string "Sep" in a file, you do it with
grep "Sep" file
This gives you all lines containing the string "Sep". But what do you do if you only want lines starting with "Sep", for example, to read all lines in your syslog regarding september? Then you need regular expressions. It works like this:
grep -E "^Sep" /var/log/messages
gives you all entries for september in your syslog. And there is much more you can do with regular expressions.