What is connectivity?
Connectivity refers to connecting devices to things, including other devices.
Controllers
Controllers mostly (if not completely) connect to consoles or PC's or whatever using either a wire, or a wireless sensor and a wireless controller. These sensors are built into their respective consoles, but if you want to connect your controller to a PC you need a wireless sensor to plug into your computer so that your controller can connect to it, as well as some kind of software that is free to download from the xbox website if you are wanting to use an xbox controller
Some people prefer their controllers to be constantly plugged into the console though, and these people get wired controllers that they can just plug into the console. With these controllers the battery will never die on you, and your controller will never mysteriously turn off while you are playing a game without much warning or reason, but there are also upsides to having a wireless controller. If you have a wireless controller then you can sit however far away from your console as you want, or even in the next room if you so please, and there is no wire to worry about so there is a 0% chance you will get tangled in the controllers wire.
LAN
LAN stands for local area network, and is how people first played Halo: Combat Evolved against each other, and therefore is why the multiplayer function was so heavily focused on in Halo 2 and was one of the reasons for Halo as a franchise being as successful as it is today. A local area network is a way for devices in one local area (like an office building) can connect to each other so that they can interact with each other.