Ever see a movie about a group of criminals pulling off heists? Payday is made to be a game where those cinematic heist scenes become playable. Running through a number of different scenarios and heists, you and up to three friends group together to get away with the most loot and rise both in power and criminal status.
The game is co-op only, the police are bots. However, heists cane be done in a multitude of ways. You can just go in, set off the alarm, and fight off the police as you try to make your get-away, but it is also possible to utilize your skills to stealth the mission and make sure police do not get alerted that you're here. It's even possible to pull certain heists so fast, that the police won't even get a chance to show up.
The objective of each stage changes per mission. In one mission, it can be the simple set-up of collect the money, get out. In another, it can be to make meth (and not make the meth lab explode while making it) and deliver it to an urban area for intel. In another you're in a mall trying to destroy as much property as you possibly can. Each mission also has randomized elements, ranging from where things are located (or how many in number), to extra sub-stages getting added on for multiple-day heists or the level layout changing.
The loot you collect gets you a net amount of spending money that you can use to buy weapons, upgrades, mods, masks, designs, skills, and more.
While you can play by yourself, like Left 4 Dead, it's best advised to play with friends. Left 4 Dead 2 is a good basis as similarities definitely, but the goal isn't specifically to kill a lot of cops, but to pull heists, and police are often avoidable if you can pull it (though there are multiple ways to accomplish any heist in question).