Supposedly, it's just surrounding the image's url with [ IMG ]URL[/ IMG ] (without the spaces) but the web host you use has to allow hotlinking (more and more rare).
For picture hosting, these days I use catbox.moe although as a privately owned, non-commercial website there's always the worry of the owner getting tired and pulling the plug on the site. Before, imgur.com used to be the obvious go-to but their recent policy changes have me pretty worried. Hosting on reddit works too (through a user post or even a post on the /r/tawawa/ sub) although you'd need an account.