You may recall that a palindrome is a string that is the same read backwards or forwards. If you ignore the spaces and the punctuation then the strings 'Madam, I'm Adam' and 'A man, a plan, a canal - Panama!' are palindromes.
(Even better: A man, a plan, a canoe, pasta, heros, rajahs, a coloratura, maps, snipe, percale, macaroni, a gag, a banana bag, a tan, a tag, a banana bag again (or a camel), a crepe, pins, Spam, a rut, a Rolo, cash, a jar, sore hats, a peon, a canal - Panama!!)
The thought-experiment swiftly persuades us that the set of palindromes over an alphabet Σ is not regular (unless Σ contains only one character of course!). After all - as you will have found by looking first at "Madam, I'm Adam" and then the two longer examples - to check whether or not a string is a palindrome one finds oneself making several passes through it, and having to compare things that are arbitrarily far apart.
Let L be the language of palindromes over {a, b}. It isn't regular, but there is no obvious bomb. However, if L were regular then so too would be the language L ∩ L(a*ba*). (The intersection of two regular languages is regular.) This new language is just the language {anbn : n ∊ ℕ}.