A competing answer, in this context, is any answer made by another user on a question where you already have an answer with a score of one or more. Note that only votes made after this criteria is met count (so if you upvote then post, it doesn't count).
As per the List of Badges on MSO:
Sportsmanship
- silver; awarded once
- Cast 100 upvotes on competing answers
- If you answer a question and earn a score of at least one, all other answers to that question are considered "competing answers"
- Upvotes you cast before you post your own answer do not count
- Deleted posts do not count
- Deleted/undone votes do not count
- Originally proposed here in June 2009; implemented October 2010.
- Sources: badge announcement blog post, Jarrod's explanation
