A birdie is when you score one under par. For example, if par is 3, you have to score 2.
When you score a birdie, the game tells you except if you do a hole in one. But if the par is 2 and you score a hole in one, you also score a birdie even if the game doesn't tell you. This counts towards the achievement.
You can quit a game before its completion and all the birdies you score will be saved (same thing for albatros).
So, if you want to boost this, the easier and quicker method is to start the Forest level, do the first hole as you want, do a hole in one on hole 2 (don't touch the right stick, set the power to 40% and shoot) and quit Forest after it records your hole in one.
Otherwise, just play normally the game as you will score a lot of birdies naturally.