Definition
A birdie in golf is a golf term to describe a score made on a golf hole where the golfer takes one stroke less than the designated par for the hole. This is often expressed as "one stroke under par" or more commonly, just "one under".
An eagle in golf is a golf term to describe a score made on a golf hole where the golfer takes two strokes less than the designated par for the hole. This is often expressed as "two strokes under par" or more commonly, just "two under".
Usage
Wow, great birdie on that last hole!
Rory Mcilroy made a magnificent eagle on the last par 5!
Origin
According to the USGA, the term "birdie" originated at Atlantic City Country Club in 1899 when Ab Smith remarked, "That was a bird of a shot!".
According to the USGA (United States Golf Association), back in 1899, there was a golf match played at Atlantic City Country Club in New Jersey where one of the players, Ab Smith remarked, "That was a bird of a shot!".
Back in that day, "bird" referred to anything excellent, so he was essentially remarking, "What an excellent shot".
Following this logic, an eagle was a more esteemed term than "birdie" while an albatross (double eagle) was even more esteemed than an eagle.