Definitely avoid overbaking - that's the most common reason for dry brownies. A lot of people bake for the full recommended time, even though every oven heats differently. A brownie should still be slightly moist and a bit jiggly in the middle when you take it out of the oven. It sounds counterintuitive, but the heat retained in the batter means it continues cooking as it cools down. So test the doneness with a toothpick a minute or two before the recommended time is up - if just a few moist crumbs stick to it, that's perfect.
Another factor is the ratio of fat to flour. Brownies need a relatively high amount of butter or oil to get that fudgy texture. If you use a very lean recipe or cut back on fat for health reasons, the result will inevitably be dry. Look at your favorite recipes and compare the fat amounts - with real, moist brownies, the oil or butter should weigh roughly the same as the flour, or even more.
Third: let the brownies cool completely before cutting into them, ideally for a few hours or overnight. While they're still warm, they seem drier because the fat hasn't solidified again yet. And store them in an airtight container - that way they'll stay nice and moist over several days instead of drying out.