For grade-level, your child only plays against player in his/her grade. There are more chances to win some trophies because each grade will have its own rankings.
For K-3, your child will play against anyone from Kindergarten to 3rd grade, but no one above 3rd grade.
Similar for other K-#. K-5 or K-6 means elementary school students, different grade numbers are due to the difference among school districts. K-8 or K-9 means elementary and middle school. K-12 is everyone.
Kids can play up in K-# tournaments, i.e., an elementary school student may choose to play in the K-12 section, rather than the K-6 section in the same tournament. Normally those kids are strong and looking for more challenges.