Guild power and prestige are ways to show who the most powerful guilds are. Guild Prestige is earned by completing Guild Challenges. The overall Might of the guild members is also factored into Guild Prestige. 

Guild Power shows the guild that is currently the most successful raiding guild. The more a guild raids, the higher their Power. Power decays over time, so if a guild stops raiding, they will start to fall down the Power leaderboard.