Operations are basically a limited time DLC thing that lasts a few months. The maps are community made and are always available from the steam workshop, or to play on community servers. The operation just lets you play them on valve official servers.
They run fairly frequently see here (image is missing : Wildfire Feb 17th 2016 - July 17th 2016):
For the first few operations paying used to allow you to access the operation maps but this has changed in the recent ones so that everyone can play them.
When you purchase the operation you get a coin in your inventory that you can level up by doing certain tasks in the operation, usually so many kills with a certain weapon on a certain map for example. In the older operations there used to also be an amount of time played or competitive wins criteria to level up your coin.
You get access to the operations case to drop for you, which in the first few weeks can usually earn you the money back you spent on the operation in the first place if you sell it on the community market
Money from the operation is split between valve and the map makers.
In the last operation there was a co-op campaign with custom maps for you and a friend to fight through which was a lot of fun.
On a further note you can buy older operation passes on the market, but this WILL NOT give you access to the expired operations, only the bronze coin for it.
Sorry I'm not so great at explaining so let me know if you want me to clear anything up more.