If you have a DSi or DSi XL, those have different carts that you have to get, just click on the appropriate section on the left sidebar.
A word of advice, these are generally meant for using homebrew (user-made DS applications), not for playing pirated games. It's illegal to have a ROM of a game you don't have the physical copy of, at least in America. Though, some people don't follow that law.... ;)
Basically for the English-language patches, some saintly person translated the Japanese version and "hacked" (for lack of a better word) the games so that the text he translated would be in English in the game.
Flash carts are wonderful little devices. Basically it's a device shaped like a normal DS cart that has a slot in it for a MicroSD card. On that MicroSD card you put the ROM file of the game (basically the game data extracted from the DS cart), and there's software on the flash cart that will play the game just like it's a regular DS game, and most flash carts have additional features like save states, built in Action Replays, and other things.
I hope this helped, let me know if you have more questions.