In brewing coffee, do I need to use filtered water and what kind of filter should I use?
There is variable here. It depends on what water your filtering and what filter you're using. Depending on your location the mineral density of water will vary. Hard water being rich in mineral and soft water being poor. The harder the water the weaker your extraction will be. Extremely soft water like distilled or reverse osmosis water will leave you with a over extracted cup of coffee.

In general it is good practice to use a carbon filter system which removes impurities but retains minerals. However if you believe your water is too hard and you're under-extracting, try using a home filtering system that removes (some) minerals.