Follow these steps:
- Go to the Showdown! website's front page and click the green button in the top right corner that says 'Play'.
- Once the client finishes loading, click the button in the top right corner that says 'Choose name', and fill out the form that appears.
- Once you've successfully chosen a name, click the settings cog in the top right corner. It is next to the name you selected.
- On the form that appears, click the second button down, which says 'Register'. Choose a password, and answer the anti-bot verification question.
- Click 'Register'. Your account is now registered.
You can register as many usernames (or 'alts', as they're commonly called) as you like.
To combat spam, many chat rooms require accounts to be at least one week old and to have completed a ranked battle before they can send messages. This applies even if your account is registered, and even if you have other alts which can send messages. If this affects your alt and the chat room you're trying to access, you will see an error message when you try to send messages.
PokemonDB itself runs a server on Pokemon Showdown, which you may access here.