Because people demanded it and Blizzard listen to their fans, Overwatch now has a competitive play mode. You can watch the video below where Jeff Kaplan from Blizzard talks about how competitive play will work in Overwatch.
Once you reach account level 25, the mode will become available via the "Play" button in the main menu. Simply select "Competitive Play" from the submenu, and you're ready to enter a match. You can join on your own or in a group, and Overwatch's matchmaking system will automatically find the best game possible for your skill level. Before you can kick off your competitive career, every new player must first complete 10 placement matches in order to obtain their initial skill rating. This rating will be a number between 1 and 100, with higher values indicating a greater skill level. Skill ratings will be visible at the beginning of every match. As you compete, your rating will increase or decrease with each win or loss based on a number of factors, including your own performance and the skill of the other players in the match.