As of July 26 2018, the Overwatch development team introduced a new feature to combat the large issue of toxicity within the game's many gamemodes. The feature is known as Endorsements, and will be detailed below.

How Do Endorsements Work?

To make it easier to understand, I'll be splitting how endorsements work into 3 categories. The current categories are:

Endorsement Rules
Endorsement Levels
Endorsement Type

Endorsement Rules

So the feature is not spammed by all players in order to boost their friends etcetera, there are restrictions to who and how often you're able to commend players with endorsements. The current rules are as follows:

You are only able to give 3 endorsements per match. This means you can give a combination of votes between your own team and the enemy team. For example, you're able to give a vote to each of your *cough* amazing healers on your team, and one to an enemy player you thought was particularly good. Any combination works - you can give your endorsements to your team, the entire enemy team, or even nobody.
You can only endorse players once per match. They aren't revertable and once you send an endorsement it is final. It's a particularly good feature considering its another preventative mesaure that reduces the amount of spam votes.
You can only endorse a player once every 12 hours. This means that if you have previously endorsed a team member that saved your ass beforehand, if you're placed with them or against them in the next game (until 12 hours later) you won't be able to commend them again, as much as you'd like to.

Endorsement Level

As you recieve endorsements for your positive behaviour, your endorsement level will go up. Recieve more and more to increase your level. (Users on Reddit have speculated the progression from level 1 to 2 is at 50)

Maintain your level by recieving additional endorsements
The colored ring around your current endorsement level shows the rough percentage on what you've been voted for. These can be seen below.

Endorsement Types

I've included some variations above. From left to right, we can see that player one has been praised as a majorly helpful, effective teammate, pretty positive and respectful and sometimes likes to be the team leader and strategise. Player 2 is once again majorly a helpful good team mate who is also pretty respectful. Player 3 overall is a good respectful sport, is relatively helpful but doesn't call shots all the time.

Ok, So How Do I Endorse People?

Excellent question. As your match is wrapping up or afterwards when you're at the main menu etcetera, you'll see an additional UI pop up at the bottom center of your screen. Click 'N' or whatever your keybind is to enter the endorsement menu.

You'll then be greeted with a menu in which to endorse your last team. If you click the arrow located at the right of the menu, you'll be able to select foes to endorse.