Your IP address will be stored, and will be used to enforce mutes and bans.
Other users will not be able to see your IP address, but moderators and server operators will.
Note that if you're on any server not operated by Pokémon Showdown or Smogon University, it is up to that server's operator to keep your IP address private. While we will deregister any server found to abuse IP addresses, we can't guarantee that they will keep them private. This is not a problem on the official server, though, where we strictly regulate who has access to your IP address.
Cookies are used to keep you logged in and remember your preferences. If you block cookies, you will still be able to use Pokémon Showdown, but will have to re-enter your username and preferences each time you open it.
Pokémon Showdown also uses Google Analytics and Google AdSense, which store their own cookies. These can also be safely blocked.
Google Analytics and Google AdSense
Pokémon Showdown uses Google Analytics and Google AdSense, which collect referrer, browser, and computer information. This information will not be shared with third parties.
Children under the age of 13 should not use Pokémon Showdown.