It depend what type of content it is. Mostly how it works is via micro/online transaction through the systems marketplace (xbox live marketplace, psn store, etc). You simply select the content you want to buy and it downloads to the system's hard drive. When a game loads up it checks for and loads any data related to the game (game saves, and dlc). It's all automatic when it comes to games loading dlc once it's been downloaded. Some games also have an income option to take you directly to the game related content in the marketplace like rock band.
[edit]: I'd like to add that unlike PC you don't have to manually manage dlc, where to save, or load it from. The systems take care of that automatically.

For Wii: it doesn't.

lol

Good to know. MK'11 will be my first experience with DLC.

Actually, Wii works best in my opinion. I have both PS3 and Wii... and I gotta say the waiting times for the PS3 are annoying as FUCK!!! With the Wii u just download and ur set, no more waiting. For the PS3 everything takes forever! Everything loads like 2000 times, damn machine forces you to update every other day... not directly caused by DLC, but still a big pain in the arse when u wanna get it. Granted, it depends on the game, but the PS3 network drives me fucking bananas its so inconvenient.

Thats the upside of the more simple OS on the wii. The shopping channel is so fucking easy and convenient.