Filters come in a wide range of sizes and shapes and some can be very expensive. They are not something that is necessary for photography, particularly in the beginning stages.
The most important filter is a UV (Ultraviolet) filter which can be bought relatively cheaply on Amazon or Ebay. A UV filter protects lenses from being scratched. When you are changing lenses to urgently get that shot and abandon your lens without lens caps, the UV filter will prevent expensive damage to your lenses.
Filters come as screw on or in a slide in plates for a variety of holders that fix to the front of the lens. Screw on filters can be a little difficult to screw on and off, where as slide in filters are quicker, however screw in filters are generally less prone to damage and scratches and don’t need extra equipment to be used.
Some specialist filters that you may come across include, polarising filters (PL), graduated filters (Grad) and neutral density (ND) Filters.
Polarising Filters can reduce reflections on glass, water and can help to darken skies.
Neutral Density Filters reduce the exposure in the case of strong ND filters, so that long shutter speeds can be achieved in daylight. Graduated filters allow a specific area (usually the sky to be reduced in exposure to match other areas (the land).