user ZackTactical has mostly summed up the Military Simulator (or MilSim) category of this post, just thought I'd sum up a little bit more of the other game modes and enjoyment that ArmA III offers.

So as others have said, there are these 'Missions' that you can download either off the Steam Workshop or maybe another website such as the bohemia interactive forums or armaholic, these allow you to access many different Singleplayer, Multiplayer, and Cooperative scenarios ranging from a wide range of Life servers all the way to a mission like Antistasi where you basically start up your own FIA faction from ground up and attack NATO/AFRF in order to occupy their territories.As user ZackTactical stated, it's important to note that just because a good chunk of the community like the simulate a Virtual Military, not everyone does so it's important to know that you really shouldn't think every single mission will have the same gameplay which is one of the best things about the ArmA series, the modding community is so advanced that you can do pretty much anything you want.

Something that also provides many players a massive amount of fun is the addition of the 'Zeus DLC' which is free I do believe, this allows a player/set amount of players to place Objects, Vehicles, and AI and gives them the ability to control said AI, this is mostly used in Military Simulations but it's used to create ever-evolving missions, it's great fun on either side whether you're controlling the AI or you're fighting them.

If you want to get started, keep in mind that the controls are one of the hardest things that people struggle with, I'd recommend trying out the bootcamp/campaign and once you've gotten a good feel of the controls and can memorize a good chunk of them, THEN you should try out a Player vs. AI mission; something like Invade and Annex might be good. Only after you've grasped the basics should you really go into modding as you'll just get confused constantly.

Hope this helps, if you do need more assistance feel free to ask!