First off, like any rule in fiction writing, "show, don't tell" can be broken - provided the breaking is done well. When it's not done well, what should be a lively, involving story can become a dry recitation.
To see how "show, don't tell" is done, look no further than Larry McMurtry's Lonesome Dove.* You'll probably be too involved with the characters and the story on the first read-through, but on a subsequent re-read it becomes apparent that many of McMurtry's characters suffer at one point or another from depression.
McMurtry can't point this out to the reader directly because of the narrative format of the book and because no one at the time the book is set recognized clinical depression for what it was. So rather than telling the reader that "Character X felt depressed that day" he'll show us that character's psychological state through the character's actions and thoughts.
It's a subtle yet effective tool. Put the reader in the mindset of a character who's drinking to dull his emotional pain, or who is so overwhelmed by the turn his life has taken that he can't bring himself to eat or to get out of bed in the morning, and you'll have conveyed the character's state of mind successfully and increased the reader's empathy with the character as well.
*Everyone should read this book anyway for its involving characters and affecting yet non-flashy prose style (be forewarned that McMurtry just loves to smite characters in awful ways).