Note: The original question here was meant to be an information-gathering survey but was clearly misunderstood and has been changed to match a useful answer to a different question.
What are tags useful for?
Straight from the tags page itself:
A tag is a keyword or label that categorizes your question with other, similar questions. Using the right tags makes it easier for others to find and answer your question.
Basically, tags are a categorisation system. What are they useful for?
To describe what kind of question is being asked. If you see a puzzle tagged riddle, you know it's going to have a riddle element to it; if you see a puzzle tagged cipher, you know it's going to have a cipher element. That way you get an idea of the approximate nature of the puzzle without having to read the whole thing.
To help increase searchability. If you want to find a particular riddle, then you don't want a bunch of cipher puzzles cluttering up your search results, so you add
[riddle] to the search terms. See also How do I search for questions with (or without) specific tags? on main meta.
To act as a 'bat signal' for experts. If I see a question tagged riddle or mathematics, I'm going to have a look at it and see whether I can apply my knowledge of riddling or maths to solve it. This is also where the concept of favourite tags comes in: an expert in geometry, say, can favourite the geometry tag so that they get the puzzles they're most interested in - and most likely to be able to answer - highlighted for them.
To warn people off the things they're not interested in. Conversely, if I see a question tagged computer-puzzle, I'm very unlikely to be able to solve it. I may give it a cursory glance, but there are so many others on this site much better suited to solving such puzzles than I. Personally I don't ignore tags, but this is where the concept of ignored tags comes in: someone who hates riddles, say, can put the riddle tag on ignore so that they don't have to be bothered by such puzzles.
For more information, see Why do we tag questions? and How do I correctly tag my questions? on main meta. Of course none of that is Puzzling-specific, but a surprising amount of it does apply to Puzzling as well as to more conventional Q&A sites.
To find out how individual tags are used, we have the tag wikis, which should ideally include usage guidance for the tag and tell you exactly what kind of questions are supposed to have that tag. These are editable by anyone (subject to community approval for those of you with less than 20k rep), so we can all take part in shaping and improving them to provide the best possible guidance for each tag.