Well, some basics:
-The Switch is both a handheld/tablet device (think Gameboy) and a console (like, say the Xbox).
You can "dock" the handheld/tablet and it will use the television. And you can take the unit out of the dock and it seamlessly transitions to being a tablet/handheld with it's own screen.
-You can take off the controller units from the devide (called "joycons"), which can function as two separate smaller controllers, depending on which game (but this is limited. For multiplayer, it is best to have two full controllers). Or you can use them combined as a traditional controller. So in summary, when you have a Switch, you can dock it, click off the controllers, and use it as you would use an Xbox.
-You can most definitely buy additional controllers and play multiplayer on a single Switch.
You have two official options: Buy an additional "classic controller", which is basically a traditional one like, say, the Xbox has. The other option is another pair of joycons (the same things included with the device), which function as a controller but can also be separated and used for some other stuff (they have Wii motion controls built in, for example, and can be clicked onto the Switch). Because they have more possibilities, I'd say a second pair of joycons is the best option. All the controllers use bluetooth, so everything is wireless.