[Update]: Blizzard is bringing back Open Queue as an alternative competitive mode.
In order to catch any potentially disastrous problems, Blizzard released Overwatch's long-awaited Role Queue into beta testing as a part of a mini competitive season. Despite some issues towards the end of the beta test, the Role Queue system appears to have been a huge success as Blizzard has now implemented it in full!
So whether you're a fan of Competitive of Quick Play, you'll be glad to hear that the days of 4DPS teams have now come to an end. Moving forward, all Competitive and Quick Play games will feature two supports, two tanks and two DPS heroes. Best of all, each role will have its own matchmaking rating, so regardless of what skill level you end up playing at, you're pretty much guaranteed to have people that are just as competent.
However, if you're not a fan of Role Queue and you're itching to return to the craziness that was the old Quick Play mode, worry not as Blizzard has also added a new Arcade mode - Quick Play Classic. Whether Quick Play Classic will rotate like No Limits, I'm not exactly sure, but on the positive side, at least you'll get extra loot boxes by playing in the Arcade!
You can read a little bit more about Role Queue, as well as check out Blizzard's answers to some of the most common Role Queue questions, in the most recent developer update. Good luck with your placement matches!