The game name Super Mario Bros. 2 has 2 meanings.
Japan edit
In Japan, a game called Super Mario Bros. 2 was released in 1986, and it was simply a sequel to Super Mario Bros.. The first American release of this game was as part of Super Mario All-Stars under the title Super Mario Bros.: The Lost Levels.
United States edit
In the United States, they thought that the Japanese Super Mario Bros. 2 would result in another video game crash because of its great difficulty. So they didn't release it for the NES. However, production began on Super Mario Bros. 3 and they needed a quick game to call Super Mario Bros. 2. Rather than simply releasing an easier version of the Japanese SMB2, they took Doki Doki Panic and added Mario characters and released it as Super Mario Bros. 2.