As the name says, tap water is the one that comes out a faucet. It has likely been disinfected with chlorine, plus filtered to remove sediments and treated with chemicals to neutralize dirt. Fluoride has also been added to prevent tooth decay.
Filtered water is tap water that has been run through filters to remove chlorine (this improves the taste) and other things such as bacteria and some chemicals. Different types of filters remove different things. Most bottled water is filtered in some way.
Purified water is water that is essentially free of microbes and chemicals. This is achieved by reverse osmosis (forcing the water through a membrane to get rid of chemicals, minerals and microbes), or distillation.
