Alright, here we go.
Fate/stay night (2006 TV anime) - F/SN has three separate "scenarios" that entirely split off from each other about 5-10% into the story, unlike most VNs that do similar stuff where you need to get 30-50% through or so. This is an adaptation of the "Fate" scenario of the original visual novel, which is the first of three that you unlock. There are also some minor things in here that are only revealed in the "Unlimited Blade Works" and "Heaven's Feel" routes in the visual novel. Usually considered to be "okay". It's not great, but it's a serviceable adaptation of the Fate scenario. It was better when it was all we had.
Unlimited Blade Works (2010 movie) - This is an adaptation of the second route, Unlimited Blade Works". It's one of the worst anime films I've ever seen. The story is condensed so far as to be almost impossible to understand and the animation falls apart after the first major fight against Berserker (which is actually pretty cool).
Unlimited Blade Works (2014 TV anime) - Another adaptation of the "Unlimited Blade Works" route, this time as a two season anime. This one's pretty good, if you're going to watch any adaptation of the original work you should make it this one.
Heaven's Feel (upcoming film trilogy) - An adaptation of the "Heaven's Feel" scenario, which is the third scenario in F/SN. HF is my favorite part of F/SN (the usual debate is "UBW vs HF" for best scenario, with some hardcore Saber lovers going with Fate), so I hope that the studio making it does a good job. The promotional poster looks
freaking great.
Fate/Zero (2011 TV anime) - This is an adaptation of a series of novels by Gen Urobuchi (Madoka Magica, Psycho-Pass, Thunderbolt Fantasy, etc). It is a prequel to Fate/stay night. Considered by much of the overseas fanbase (can't speak for the Japanese one) to be the best that Fate has to offer as an anime and often as the best Fate-related work overall.
Fate/kaleid liner PRISMA☆ILLYA (I don't care enough what year(s) this is in to look it up) - A TV anime adaptation of a parody spinoff of F/SN starring Illya where she becomes a magical girl and does magical girl stuff. There's a ton of fanservice in it for lolicons so I've never bothered with it, you probably shouldn't either.
Fate/Prototype (2011 OVA) - A 12 minute special that animates some scenes from the original planned version of Fate/stay night, which underwent extremely heavy revision (it's almost unrecognizable) before becoming what we know today.
Fate/Grand Order: First Order (2016 special) - I haven't watched or paid attention to this one, but to my understanding it's just an adaptation of the opening chapter(s) of Grand Order.
Carnival Phantasm (2011 OVA) - An adaptation of a gag manga featuring characters from Fate/stay night and Tsukihime (one of the company's other major works). It's really funny, but best not to watch if you're unfamiliar with the original works. You should at least watch one of the full adaptations of one of the routes of F/SN before watching this.