The sole intention of this database is to provide a collection of League of Legends resources for fanbase sites and applications.
League of Legends - Champion Database uses unlicense
League of Legends, characters, images and all other copyrighted content related to League of Legends are the property of Riot Games, Inc. All rights reserved.
Riot Games directly allows 3rd party websites to use their art and champion skins.
From their FAQ obtained by emailing fansites@riotgames.com:
Q: Can I use your art/logo/champion skins on my website? A: As long as you aren't charging people for access to your website, you may use any copyrighted assets we freely distribute in the game client. This includes all champion skin splash images.
The exception: you can't use the Riot Games logo or other trademarks under any circumstances.