I got drunk and woke up with a tattoo - what do I do with it?

Tattoos, barring one exception detailed below, are completely cosmetic and serve no in-game purpose other than to display in your profile and be yet another thing you can collect.

There are many tattoos and they are unlocked through various methods. The most common ways are through collecting Outfits and Ascending. Once you qualify for one, go to the The Dirt-Walled Hovel of the Pretentious Artist

The Demon Tattoo is currently the only tattoo that has an effect on gameplay. It will allow you to negate the damage of the Burning, Man effect.

To view your list of unlocked tattoos go to Options > Account Options > Manage/View Your Tattoos.

The particular tattoo that you unlocked is the Martini tattoo, which has a 1 in 4 chance of unlocking whenever you consume a Martini.