Discord Bot List (top.gg) - Most popular Discord bot list website, has a large directory of bots. Has very strict rules for bot developers, such as no nsfw commands (urban dictionary counts as one) and no banned words in the bot name (f.ex "loli").
Discord Bots (bots.gg) - Large bot directory with many newly-submitted bots. This directory allows bots with NSFW commands.
Carbonitex - This used to be the most popular Discord bot list website. It has a nice database of good Discord bots, but unfortunately the website doesn't add new bots.
botlist.space - Another great website to find the bots you want. This website allows bots with NSFW commands.
Discord Bot List - A small website with bots. It's hard to use when you want to find bots for your server, but feel free to submit your Discord bots here.
Divine Discord Bot List (CLOSED) - A small list of bots. Another place to submit your Discord bots, it also has it's own API wrappers.
If the description is inaccurate, a link is broken or if there's a website you want me to add, feel free to contact me on Twitter @hug_onun.