The campaigns story is that in 2025 US Navy Seal Commander David 'Section' Mason is tasked with taking down a known terrorist by the name of Raul Menendez. During his attempts, Menendez manages to gain control of the entire unmanned drone fleet, of the US, in order to launch mass atacks on cities across the world. Meanwhile Sergeant Frank Woods, now retired, recalls his experiences in the 80's with Alex Mason, son of David Mason.
The Multiplayer has no story, essentially just being a sandbox of game modes you can play with friends, strangers, or bots.
Zombies, my favorite mode, consist of several maps including tranzit, mob of the dead, die rise, etc... each map has its own background lore that does eventually connect together, but you have to look through all extras and possible easter eggs to fully understand it.