Mithril is an incredibly strong, light, and beautiful metal. A coat of mail made out of mithril is strong enough to block a stray spear or knife (both of which Frodo is unlucky enough to encounter during his travels). Frodo’s mithril coat happens to come from Bilbo, who passes it on to his nephew in Rivendell. Bilbo himself originally received the mithril coat from the hoard of the dragon Smaug after the Dwarves retook Erebor, the Lonely Mountain, at the end of The Hobbit.
So, that’s what mithril is. But what does it symbolize? Well, if we look closely, it appears to represent hidden (Hobbit) potential. When Bilbo passes his mithril coat on to Frodo, he tells him to keep the armor under his clothes. In other words, Bilbo encourages Frodo to let himself be underestimated: it's always easier to defeat people who don't take you seriously. When Frodo first wears his mithril-coat (again, in secret), Gandalf looks at him penetratingly and comments, "You take after Bilbo [...] There is more about you than meets the eye, as I said of him long ago" (2.5.61). The mithril-coat reminds us not to judge a book – or a Hobbit – by its cover. Underneath a lowly Hobbit's ordinary exterior, you may find a mithril-coat waiting to surprise you.