Mezes are basically appetizers. Many times in Turkish restaurants the waiters will carry out a plate or there is a display case where you can see all the different options and you'll choose a few before the main course. Personally I love to have just mezes as dinner because they are light and mostly vegetarian.
The closest thing to ezme that most people know is probably Gazpacho, the Spanish-style cold tomato soup. Like Gazpacho, you eat this cold, as with a lot of mezes. I cut the vegetables into smaller pieces and throw everything into a food processor. You can dip bread into it or just eat it alone like I do.
3 green peppers (the kind in the picture below, not bell peppers, but if that's all you can find then use bell peppers)
3 cloves garlic
1 tablespoon red pepper paste
2 tablespoons pomegranate sauce (hard to find so if you don't have it, use lemon juice)
2 tablespoons olive oil