Which when for what?
The question I see asked most about filters is "I read that light pollution filters don't work; should I buy one?" The answer to that very specific question is no, now go away.
For DSLR/Mirrorless buy Clip-in or 77mm screw on filters. Never buy screw on filters smaller than your biggest lens; then buy step-up rings to fit the big filters on your small lenses.
For Dedicated Astro I recommend a 2" filter tray. You can get everything from a single filter to a 7 filter electronically controlled wheel. Just do 2" because you don't want to save a little money now with 1.25" then have to re-buy all your filters and holder when you get a camera with a bigger sensor.
To clarify, in the recommendations below if it says Clip, 77mm, or Square those filters are for DSLR/Mirrorless, the filters listed as 2" are only for Dedicated Astro cameras.
Light Pollution filters are intended to remove the yellow of sodium vapor light from light pollution. That is all and they do that very well. The problem is that it still leaves all that white light coming from light pollution. Do not expect an LPS to create darkness out of bright light or you will be disappointed and be one of those that say they don't work. These filters are good for wide angle astro-landscape when you want to remove a yellow cast and color correct in camera.
Recommended: ICE LiPo 77mm or Optolong L-Pro 2"
Didymium filters are intended to protect welders eyes and do that very well; wait, what? They block all light by 3 or 4 stops. Add an extra stop for yellow, and remove one for red. Odd as it seems this is my preferred filter for Milky Way shots.
Recommended: Hoya Starscape 77mm or NiSi Natural Night square or Baader Moon and Skyglow 2".
Ultra High Contrast filters straddle the line between filters designed to only block specific frequencies (light pollution) and those designed to allow specific frequencies (nebular gases). I would recommend these over LPS filters when shooting deep space.
Recommended: Optolong UHC EOS Clip or Optolong UHC Nikon Clip or SVBony UHC 2" or Baader UHC-S 2"
Multi-Narrowband filters. Remember when I told you Light Pollution filters are not intended to block light, just color; well Narrowband filters are meant to block light. The problem is you don't want to block the light if you are shooting Milky Way. Galaxies are best captured using the entire visible light spectrum. But nebulae, now we can do something about them since they can look good using only very specific colors.
Recommended: SVC Duo Clip or Optolong L-eNhance EOS Clip or Optolong L-eNhance 2"
The I'm Made of Money Recommendation: Radian Triad Ultra 2", they also make a 65mm Square version
UV/IR filters are not needed for DSLR/Mirrorless cameras as they are builtin; although If the infrared blocking happens to drop below 665nm it causes problems with Ha and is why they are sometimes "modified" out. Even my dedicated astro camera has one built in, but the cutoff isn't as sharp as I would like and allows for bloated stars. If you are using a modified or a dedicated astro and your stars, even at the best focus, are bigger than they should be then you may need a cutoff filter.
Recommended 2": SVBony or Baader
Broadband Filters = designed to block only specific frequencies
LPS or CLS = Light Pollution Suppression
These are the light pollution filters you have heard about.
Mercury Vapor = 400nm, 550nm, or 575nm (anywhere from Blue to Yellow, but gives everything a Greenish tint)
Sodium Vapor = 550nm - 650nm (Yellow)
Didymium or Neodymium = Light Pollution Suppression + Enhance Red
Haven't found usable specs for this but from my experience they give the effect of blocking Sodium Vapor and brightening Ha at the cost of 3 stops of light.
UV/IR Cut = Non-Visible Light Cutoff
Needed for modified DSLRs and dedicated Astro cameras.
Ultraviolet = < 400nm
Infrared = > 700nm
UHC = Ultra High Contrast
Straddles the line between between blocking and allowing. Between Light Pollution and a Quad-band filter.
Hb - OIII & Ha - IR = allowed so a very wide dual-band
Narrowband Filters = designed to only allow specific frequencies
LRGB = Visible Light
Mainly used with mono cameras to produce color images.
L = Luminance = 400nm-700nm (all visible light)
R = Red = 600nm – 700nm
G = Green = 500nm – 600nm
B = Blue = 400nm – 500nm
Single Narrowband = Only 3nm - 12nm around a specific frequency
Mainly used with mono cameras to enhance nebulae as these are the gases of which they are mostly composed.
SII = Sulfur = 672nm (Deep Red)
Ha = Hydrogen-alpha = 656nm (Red)
OIII = Oxygen = 496nm - 501nm (Blue/Green)
Hb = Hydrogen-beta = 486nm (Blue)
Multi Narrowband = could be referred to as Duo, Tri, or even Quad band
Mainly used with color cameras to enhance nebulae as these are the gases of which they are mostly composed.
Ha & OIII = dual-band (STC Duo, Optolong L-eXtreme)
Ha, Hb, OIII = tri-band (Optolong L-eNhance)
Ha, Hb, OIII, SII = quad-band (Radian Triad Ultra)