It's one of the Internet's leading sources of skub, so we'll try to confine ourselves to a few simple facts.
In essence, it exists to catalog the various recurring patterns in stories, characters, and general entertainment production and consumption. As such, it can be quite an addictive read (insert "TVTropes ruined my life" joke here), to the point where it's easy to lose track of several consecutive afternoons even after you discover their far briefer "Laconic" sections.
It attracts a great quantity of weeaboos, neckbeards, fanboys, fangirls, fanhermaphrodites, other weirdos, and just plain losers as editors - and it shows frequently in the general writing style and the types of pages included. It's a great place, in theory, to go to find ideas for your next campaign - but don't overdo it, since that's only in theory. Unfortunately, besides being a slippery slope of a time sink, TVTropes is the kind of site that lends itself to categorization for categorization's sake also like a lot of other wikis - but it's amplified so far that its users can no longer see any kind of media, or anything else in the worst cases, as anything other than a collection of archetypes and cliches. Poorly written examples and argumentative sperging over the smallest and most seemingly insignificant of minutiae abound, and that's not even getting into the times where they try to force tropes into applying to the real world with predictably disastrous results. And then there's the horror stories of the old "This Troper" section...
Beyond that, it's worth mentioning here because, coverage of all types of media aside, it has had a definite influence on fandom and internet culture in general, being a product of such. It's also prone to in-jokes that occasionally leak out onto the wider Internet (sometimes known as a TVTropism). As such, it'll probably show up at your table, either directly mentioned by someone trying to seem more learned and experienced than they really are, or by other players using certain terms from it. You may or may not find particular phrases little here and there in this wiki's articles.
No matter how much we might deny any connection between ourselves and them, they have an entire section of their Warhammer 40,000 Fanfic Recs page devoted to works coming from 4chan (including many stored here).
Your Approval Fills Me With Shame *BLAM*
To speak of it any further is to invite madness.
- You can find it here. Do not click that link unless you are prepared to accept the consequences. To protect people from butthurt, they sectioned off several other areas from the main page, including Your Mileage May Vary/YMMV, Flame Bait, Wild Mass Guessing, and so on. If you haven't whitelisted their dumb-ass invasive and history-rewriting ads, they'll bug you about it each time you open a page. Remember: it can be very addictive, it may make you see entertainment as only a series of tropes, and it may cause you to start using their page names in normal conversation like an idiot.
- The most notable fork is found here; All the Tropes was formed due to censorship issues on TV Tropes - mainly a past problem with advertisers and the tendency for some of the site's moderators to (ab)use the banhammer for selfish reasons. Fortunately, while they brought back the PROMOTIONS-style material, they left out Troper Tales and other cancerous friends. However, it is also painfully out of date and needs a Titan-load of work done to catch up with most works made after 2012 - a rather obvious downside to being less popular. Thankfully, the work is coming along nicely; in addition, the staff are a lot more helpful than those on TV Tropes, you don't get the banhammer for expressing "wrongthink", and you don't need a username to edit. Also, they have no ads. The main admin actually wrote several GURPS books for third edition, plus the 4th edition GURPS Basic Set: Characters.