Death by Genre Savviness: The player, assuming that he/she is familiar with the general gameplay of Super Mario Bros. 1, will die over and over again precisely because the entire game is devoted to subverting that Genre Savviness and killing you. For example, that pipe that should lead to a secret area? Kills you. That star that should make you invincible? Kills you. That invisible block that should contain a 1-up mushroom? Contains poison mushrooms and kills you.
The Dev Team Thinks of Everything: Nearly every possible route through the levels was foreseen by the developers, and then all but one of those routes were all filled with unavoidable death traps to limit the player to using only a single precise route. For example, in the fourth level, there's a place where you needs to jump over a lava pit. Jump too close to the ledge? You hit an invisible block. Try to jump onto that invisible block and then jump over the lava pit? The invisible cloud kills you.
Didn't See That Coming: Many traps are well-hidden and come out of nowhere, such as jumping off a ledge, hitting an invisible block, and falling to your death.
Emoticon: Syobon, Buun, and Kuma are in fact 2ch emoticons given life. See Memetic Mutation entry below.
Escort Mission: In order to beat level 5, you have to guide a spiny enemy that jumps when you do through an upper path, while you travel along the lower path, which is naturally filled with traps. You need him to eat a mushroom and break through a wall.
Everything Trying to Kill You: Even the clouds want you dead. Even the stars want you dead. Even the pipes want you dead. Even the floor wants you dead!
Expy: Given that this is a Super Mario Bros. 1 parody, most enemies are expies of Mario enemies such as Goombas, Spinies, Thwhomps, and Bowser himself. Generally, they're cat-themed.
Fake Difficulty: A lot of the difficulty comes from all of the traps. However, the game does require quite a bit of skill and patience, so it's definitely not all fake difficulty.
Fan Sequel: There are many "sequels" to this game out there, and they are even harder than the original one.
Invisible Block: Placed in the most frustrating areas and will kill you, naturally.
Red Herring: The Bowser Expy and poison mushroom at the end of the last level. Either one kills you, but eventually you'll figure out that you're supposed to use the mushroom to kill "Bowser" instead... but, if you do so, all that does is lead you straight into one last unavoidable Kaizo Trap in the form of a giant chicken man standing in for Toad. To beat the game, forget about "Bowser" and the poison mushroom and instead let the throwing enemy carry you above the level, allowing you to bypass "Bowser" and "Toad" and find the real ending.