In the PvP arena you are matched against players and compete with them for a gem reward. You have one week to collect as many stars as possible to be promoted to a higher league. Top three players are promoted to a higher league when the week ends while bottom three players are demoted to a lower league. Make sure to collect at least some stars each week so that you won’t be demoted!
The score depends on the level of the birds; the higher the level the more points you'll get. You can level up your birds by finding feathers; you'll notice that it is easier to compete against others once you have higher level birds. You can see the level of your birds and how many feathers are needed to raise their levels in the nest.