In brewing coffee, is it important to use filtered water? If so, why?
Is there a particular type of filter that is most effective for water to be used in brewing coffee?
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.
In addition to Charlotte's cook answer I'd note that heating / boiling water removes most of the common volatile compounds such as chlorine (not minerals) that may affect taste. So your boiled water would be more neutral in flavour than the same water straight from the tap.
So depending on your water supply, there is often less need to filter for making coffee than cold beverages.