Fish Tank Pollution?

Hi! We have a 10 gallon fish tank that currently holds three tetras (one black, two rainbow). For the past few months, the tank has been forming a brown fuzzy coating inside it--it coats everything in the tank, and is difficult to clean off. Water changes do not remove it, and it clings to the pebbles and plants. Recently, I broke down the entire tank, placed the fish in a bowl, and scrubbed the entire tank and its contents (including the pebbles). I re-embled the tank, let it "settle" for a day, then placed the fish back in. But the brown fuzzy slime is back!! What is it, and how do I get rid of it for good? And would getting an algae-eating fish help?