I read the math FAQ and searched meta.math. Some badges are awarded for answering a question with score 25 (or other number) or more. What is a question's "score"?
The score of a question or answer is the number displayed on the upper left, that indicates the total number of upvotes it received minus the total number of downvotes it received. For example, this question has received 219 upvotes and 28 downvotes, giving it its current score of 191. Since its score is 100 or greater, its asker received a Great Question badge. Similarly with answers: this answer to the question received 324 upvotes and 1 downvote, giving it its current score of 323. Since its score is 100 or greater, its poster received a Great Answer badge.
By the way, one gains the ability to see the separate upvote and downvote counts with a reputation of at least 1000.