The dirty martini has a pleasant saltiness that is fascinating against a gin and vermouth background. "Dirty" simply refers to the addition of olive juice or brine. It's a classic cocktail that is very easy to mix up and one of the most popular variations on the original gin martini.
You can make this drink as dirty as you like by pouring the olive juice to suit your taste. It may take a few rounds to find a perfect balance for you, but the experiments are fun. As you explore different brands of gin, you will want to make adjustments as well.
The key to making a great dirty martini is to use high-end gin (or vodka, if you prefer) and vermouth and to add just enough olive juice to lightly flavor it. There is a big difference between making it "filthy" rather than just "dirty," so take it easy at first.