grep is just one of a family of commands that use regular expressions. egrep provides some more sophisticated regular expression features (and there is also an fgrep that does not do regular expression searching at all, ostensibly to be faster for simple substring searching).
The major regular expression extensions provided by egrep (and also available in some other tools that handle regular expressions) are:
There are also some commonly-used options to the grep-family commands:
As usual, man grep will give you a much more detailed listing of available options.