>>956899My Anime List. It's the most popular anime and manga database that also doubles as a list service. You can create an account and then added series to your list from their database, sort them by "completed", "dropped", "plan to watch", "currently watching" etc, give them ratings, track specific start and end dates for watching, and so on. MAL isn't the only database or list service out there, there's also Anilist, Hummingbird, a few others, it's just the one with the biggest community. That only really matters if you care about stuff like user ratings and user reviews, since MAL also lets people post reviews for anime and manga, and compiles a whole bunch of statistics about shows (but the main one people pay attention to is average user rating).
You might wonder why people keep and maintain lists of what they've watched and are watching, but if you are watching a lot of anime at the same time (which happens when you watch anime seasonally), it's a good way to keep track of which episodes you're on for which shows. Some people also just like to track everything they've watched.
Personally I don't use MAL, I had one years ago, but ditched it for Anilist since it integrates better with Taiga, which is a desktop application that connects to one the databases and your profile and then syncs your local watching activity to that list automatically. It does this because it can detect what files you're watching in certain directories on your media player, then if it finds a matching title on your list, it will update watching information. This means I don't actually have to visit Anilist's website to update my list, Taiga does it for me. It's a nifty little app that's only really useful if you watch a lot of anime though. Pic related is what Tagia's UI looks like.