Redirects are used to forward the user from one page on the wiki to another page.
We use redirects to connect logical search phrases to helpful places. They don't cost us anything. There are a few exceptions, but unless one redirects to a place to where someone wouldn't want to go, there's no downside to having them. In such a case where a phrase could have multiple meanings, a disambiguation page is used.
For example, let's take a look at the search phrase "luigis mansion 2". Without a proper redirect, users are taken to this results page. MediaWiki's built-in search is not great, and redirects allow us to make up for that by applying a human touch to point users in the right direction. If there's even a small chance that a redirect will help someone, it's not useless.
Redirects are only unnecessary when (1) they are completely nonsensical, (2) they are too general, (3) they overlap each other, and (4) the search function takes the user to the correct page without said redirect. This generally means that pure capitalization redirects are unnecessary, but cases should be tested using an all-lowercase search for the page title. "Paper mario 3ds" may seem unnecessary because you'd think the search function would pick up on the "Paper Mario 3DS" redirect when the user searches "paper mario 3ds", but uppercase acronyms (whether they contain numbers or not) are not currently detected by MediaWiki when searching in lowercase. This is why we test by searching!
(1) We don't need the redirect Game with luigi and ghosts 2 because it's too silly.
(2) We don't want the redirect First mario game because we're not positive what page the user is looking for (you might say Donkey Kong (game), but maybe they're actually trying to find Super Mario Bros.), so the search page is the best place for them to be.
(3) We don't require the redirect Mario galaxy because the redirect Mario Galaxy already takes care of users searching for the phrase "mario galaxy". As a general rule, the case of the redirect should match the source. Due to MediaWiki being picky, two redirects must exist for each text phrase when we're dealing with acronyms (any group of two or more characters where all the letters are uppercase, whether they contain numbers or not). The first should use the proper capitalization (e.g. NSMB Wii, Paper Mario 3DS) and the second must make all letters lowercase (e.g. Nsmb wii, Paper mario 3ds).
(4) We don't require the redirect Super mario galaxy or Super Mario galaxy because entering "super mario galaxy" in the search field takes the user to Super Mario Galaxy without any redirects.
Spelling mistake redirects are a grey area. The general rule is: don't think about them too much. Don't worry about tagging them for deletion unless they break another rule because there's really no point, and don't go off creating as many spelling mistake redirects that you can think of. We suggest reasonable mistakes aimed at more-popular subjects, which is subjective but not 100% subjective. Porplemontage (talk) creates them based on our top search phrases, but everyone else has to either make some guesses or just not worry about them.
Other types of redirects
Template, Category, and File redirects are discouraged. In the rare case that one of these types of pages is moved, it is necessary to update all links to the new name and delete the redirect page.
Linking to redirects
Linking to redirect pages is discouraged, so always try to link to the actual page name and use the piped-formatting to correct the name if necessary.