I think, together with ND filers, this is the only type of filter I now use with my digital cameras. It's the only filter that cannot be reproduced in post processing.
The polarizing filter (also known as a Pol filter) can actually do more than darken a blue sky. It can take away reflections from windows or dampen reflections from wet leaves. It also lengthens your shutter speed by up to 2 stops. Those are things that you cannot really do after the image has been taken.
You really only have two sorts of pol filters; the Linear and the Circular. The latter is just two linear filters stacked upon each other.
The circular is much more expensive than the linear and is the one that has to be used when using DSLR cameras. The reason for this is the prism inside the DSLR. If you use a linear then it could cause havoc with your focusing and also your metering systems.
However, with the introduction of the mirrorless cameras (which don't have a prism) linear pol filters can be used. This means you can save quite a bit of money with these filters alone. This doesn't mean that the cheapest pol filters should be bought, those can, and do, cause colour shifts when used. The best and more expensive filters do not cause image degradation when being used.
Care has to be taken when purchasing these filters. I personally only buy these filters at a reputable brick & mortar store (when available) or online when I know they have a good return policy. Reason being that fake filters are on the market and do cause a lot of problems, especially for the new photographer who lacks experience with this sort of thing. When you receive a new filter test it as soon as possible and when you do experience a problem with it, just return it for a refund. Then try another, preferably from another retailer. Once bitten, twice shy as they say.
Since I only use mirrorless cameras these days I use linear pol filters most of the time. If I have a circular pol filter in a certain size then I'll use that one too. It doesn't really matter with a mirrorless camera. Both can be used. I don't like using step-up rings so I have a filter for most of the common sized filter threads. Thankfully the range isn't all that great and manufacturers keep their lenses within this range of filter sizes.
Normal filter sizes:
46mm,49mm,52mm,55mm,58mm,62mm,67mm,72mm and 77mm.
There are a few sizes outside of that range but not that many. If you have filters for those sizes then you have a filter for most lenses on the market today.
If you have circular pol filters and have switched to a mirrorless system, those circ pol filters can still be used with your new system. Just when switch to a DSLR system from a mirrorless is when you would have to purchase new filters. That can be pretty expensive as good quality circ polarizers go for €200 upwards. Expect to pay at least €200 for one filter, especially for the larger sized ones.
It's not very often that I use one of theses polarizers at full strength, especially with wide-angled lenses. I have experienced myself the affect of this when the centre of the image in the viewfinder is very dark but both the left and right sides are much lighter and brighter. A blue sky is excellent for seeing this and is a good way to test if the pol filter is in fact genuine and that it works. When this happens the resulting image looks very unnatural and is caused by the angle of the light that is being polarized.
Ensure you set your white balance to the correct setting. When shooting in daylight conditions then your white balance setting should also be set to daylight. Cloudy conditions? Then set your white balance to cloudy. Never use auto white balance when using a polarizer because the camera will compensate for the polarizing effect. Make it a habit of checking this when mounting the polarizer on the lens. It happens to us all that we sometimes forget to change a setting when conditions change.
Only set the polarizer to the required strength for the scene. You don’t have to always set it to the maximum effect. It may not be required and the image may seem unnatural. Use a polarizer with a wide-angle lens at your peril. Watch the edges in the viewfinder. I personally use it judiciously on wide-angle lenses these days.
It’s true that seeing the effect of a polarizer is easier when using a DSLR rather than a mirrorless camera. However, if the photographer can’t see any difference whatsoever then the polarizer may be a fake (which isn’t all that rare these days) or the camera could be auto white balance.
Another trick with mirrorless cameras is to half press the shutter button and then turn the pol filter. You should then see some difference in the EVF. On mirrorless cameras you also have the option of seeing the effects changing the exposure in the EVF. Make sure this is turned ON. If not you will not see any difference.
Using these filters should be practiced (it takes time and experience to decide just how much polarization is required for a certain scene). So practice, practice, practice.
Some very good manufacturers are Lee, B +W, Heliopan and Rodenstock. Be aware however that these filters can cost you an arm and a leg. More affordable manufacturers include Hoya and Marumi.
A general here is like everything else: Buy the best you can afford.
Following are a few images of different pol filters. It may help you in some way.
B + W 67mm circular pol filter
Heliopan 52mm circular polarizing filter
A stack of Hoya linear polarizing filters
Liner pol filter in packaging
This is how they look
This is how they look
If you look through the polarizer in the image you can see some of the polarizing effect
Here's a link to a good explanation of polarizer useage.