Android - How to check what apps are running in Android 6.0?

This is now in the Developer options area of your device settings, under the "Running Services" menu item:

Note that if you cannot see the Developer options at the bottom of your settings, it is because they are hidden by default and need to be manually enabled.

Thanks eldarerathis

This is strictly running services and not apps.

...To actually find and stop running apps you can go to Settings, Memory, "Memory used by apps" (ADD: Strictly apps that have been running within the last 3 hours, and ONLY those using more than a minimum limit of RAM)

When you click one, the vertical ellipsis top right gives you easy access to actually STOP the app.

Note also that parallel to the "Running Services" described, there is an "Inactive apps" at the bottom of the developer options. And you should try to inactivate an app instead of actually stopping it. If it is well behaved, this will often be almost as good, and is more clean than giving it the axe with stop.