There are basically two types of filters:
- Screw-on filters.
- Filter holders with rectangular filters.
Both systems have their pros and cons, which I will address in a moment. Do note, however, that both types offer ND, gradual ND, color, and polarized filters, which make up for most of the filters anyone will need (and the list probably is not even complete).
Note that both (might) offer something between uncoated and multiple-coated filters. Coating should prevent ghosting reflections, but whether it does a good job at this or not depends on how well it is done. More expensive filters usually offer superb coating, though it is not a bad idea to lookup tests for the particular filter you want to buy.
Lenses usually offer a thread on the very front of their body. To use the filter, simply screw it onto the thread of the lens.
To save money, it is usual to buy your (effect) filters only in the size that your (most wanted or already existing) lens with the largest thread needs and then use step-up-adapters for all other lenses. E.g. if you want to buy a Canon 24-70mm f/2.8 L USM II with its 82mm filter thread at some point, you would buy 82mm filters and step-down-adapters for all your other lenses. This of course could lead to mechanical vignetting by (badly designed) step-down filters and also, it will prevent you from using most (if not all) bayonet lens hoods without screwing on/off the filter every time.
Filters that need to be rotated - e.g. gradual NDs and polarizers - are made so that they can be rotated even when screwed on the lens tightly.
Pros of screw-on filters:
- Lenses with weather sealing usually need an additional front filter screwed on for full sealing.
- If "native" sizes are used, they are not taking up much space when used.
- Do not need much space in your backpack.
- Wide price range, but usually far less expensive than sheet filters.
Cons of screw-on filters:
Photo stolen from bhphotovideo.com
- Not always compatible with lens hoods.
- Can lead to mechanical vignetting, especially when stacked.
- Sometimes get stuck on the lens - usually, that can be addressed with using more torque (e.g. via a filter wrench).
- For best fit, you would need every filter once per thread size.
- Not all lenses (especially ultra-wide angles and tele primes) offer threads for these filters. (Most of the more expensive examples offer a slot at the back of the lens for this, however.)
Photo stolen from ephotozine.com
These are rectangular sheets of glass, fitted to a filter holder. The filter holder can be mounted onto the lens either via the lens's thread mentioned earlier, via the lens's bayonett, or via mounting it on the lens's tube.
They come from different manufacturers and in different sizes, though 90mm, 100mm, and 150mm seem to be widely supported sizes. Which size you need usually depends on how wide your lenses get - 150mm usually is only needed for ultra-wide lenses.
Pros of sheet filters:
- Filter holders are usually freely rotatable and the sheets can be moved up and down, so e.g. adjusting a gradual ND filter is easier/more feasible than with screw-on filters.
- One size fits all: Your 100mm filters from brand XYZ will fit the 100mm filter holder from brand ABC, and both will (usually, see "cons") fit lenses from virtually any brand.
- At least 150mm filters will usually work with even the most extreme ultra-wide lenses. (Well, apart from monsters like the Fisheye-Nikkor 6mm f/2.8.)
Cons of sheet filters:
- Relatively expensive.
- Blocking stray light requires more than the average lens hood.
- Usually do not offer any shielding from environmental factors, e.g. rain.
- It might be hard or unfeasible to fit the filter holder on some more unusual lenses.
Which system you buy into is entirely up to you. Most amateurs and semi-professionals stick to screw-on filters, mostly for budgetary reasons (or so I think).
I myself am (at the moment) going for the approach of buying high-quality screw-on filters in the size of the biggest thread any of my anticipated lenses offers, buying step-up-adapters for all other lenses and a set of screw-on lens hoods for the diameter of the filters, thus offering some protection from stray light without having to fiddle with bayonet lens hoods.
But that is my personal approach, which is based entirely on my personal situation, and therefore, please treat it as opinion, not as a factual statement that this is the best approach.