An eagle is for finishing the hole two under par. In the top right corner the game tells you which stroke you're on and what you'll get if you make it that shot. You can't get this on a hole that is par 3, because two under par would be a hole-in-one instead. Here's a list of the courses and which holes you can earn an eagle on based on par score.
Santa's Village Fairway: None of them, they're all par 3
Golden Apple Vineyards: All of them
Pirate's Refuge: Hole 2
Medieval Town Fairways: Holes 1 and 3
The Industrial Zone: Holes 2 and 4
Landmark Lanes: Holes 2, 3, and 4
Here are some videos to give you an idea of how to get an eagle, I do make some mistakes, but they show you what you can do. I'll add more if I can get eagles on more of the holes.
Golden Apple Vineyards - Hole 1
Golden Apple Vineyards - Hole 2
Pirate's Refuge - Hole 2
Medieval Town Fairways - Hole 1
Medieval Town Fairways - Hole 3
Landmark Lanes - Hole 3