Regular expressions are the Swiss Army Chain-saw for string manipulations and no one disputes that Perl really shines here. I would say they are almost indispensible when working with text data. Regular expression engines have recently been added to the core Java platform and the Oracle database. I know that I miss them in my Parrot experiments (Parrot already has a working regex engine, with all the low-level opcodes required for that, but no compiler to target it). While Perl did not invent regular expressions, it has made them popular, and Perl's implementation has become the gold standard of the field (the most popular C library for the task is even called pcre -- Perl Compatible Regular Expressions).

Regular expressions are a (rather cryptic) language unto themselves, and I am not going to describe them here in detail. Have a look at the Quick Start Tutorial or the regular Tutorial.

The funny thing is, now that everyone has copied Perl's syntax for regular expressions, Perl 6 is going to throw it overboard and start anew. See Exegesis 5.