What is a filter? A filter is like sunglasses for your lens. They protect your lens and allow for less light to come in, allowing for your photos to have more contrast, polarization, longer exposures, and with sometimes special effects.
I used to think that filters were useless and they didn't do anything to your pictures, just a way for a company to make money. I mostly thought this because I bought cheap filters from Ritz camera that were poor quality and well, didn't have any impact to my photos. I later on purchased a ND (Neutral Density) filter and a CPL (Circular Polarizer), both from Hoya. The filters changed every single one of my photos making a huge positive impact!
One of the most commonly purchased filters is the UV (ultra violet) filter which is only used to protect your lens.
CPL is the filter you should always have if you are shooting landscape. A CPL is best used when facing 90 degrees to the sun. Make a L with your index finger and thumb and point your index finger toward the sun... Now wherever your thumb is facing will give you the greatest impact while using your CPL. CPL gives you deep and strong contrast while giving your sky a nice rich deep blue color. Also using a CPL will block out light giving you a longer exposure time to let you have those pictures of moving water, people, etc. Suggested CPL's are HOYA, B+W, Heliopan, LEE, and Singh-Ray. Having a good CPL will really make your pictures stand out, so invest in a good one!