Nintendo was founded in 1889 originally to produce cards. Later, it began making toys. Starting in 1975, they started making video games.
First: Not really a logo, but on a background is the words Licensed by Nintendo. On some games, Licensed by Nintendo of America, Inc. appears instead of Licensed by Nintendo.
Second: On a black background, we see - Nintendo - in it's font without the round shape. Below it is Presents".
Third: Same as the Nexoft logo used in Penguin Wars, but with the corporate Nintendo logo, but without the oblong shape.
Fourth: On a black background, we see the normal Nintendo logo but the round shape is stretched out a bit. On top the text is a gold Mario coin, that shines once. On Super Mario Kart, Yoshi's Safari, Super Punch-Out!! (this game has the logo in silver), and Stunt Race FX (this game has the logo in pink), there's no gold Mario coin.
Fifth: Depending on the variant: Killer Instinct: TBA Mario Kart 64: We start off with a side view of the Nintendo logo. The logo is gold. Suddenly, it starts spinning around, getting faster and faster. As it does this, a trail of pieces of the logo shows up, and the logo remains this way as it fades to white. Goldeneye 007: The Nintendo logo, in silver, zooms in suddenly and spins around at a slow and steady pace. Perfect Dark: The silver Nintendo logo tilted downwards in the darkness appears and faces the viewer while the light brightens up to make the logo visible as it slowly turns and becomes dark again. On most Nintendo games: The Nintendo logo is red and is on a black or white background. Starting in 2006, the logo is now grey. Donkey Kong 64: Same as the previous variant, except it suddenly appears. On the Japanese release of the game, the Nintendo logo is silver instead of red. Most other games have the logo still, along with being silent.
Sixth: On a black background, we see grey smoke rising before a red skeleton with its eyes lit green fades in, and looks to the camera with its mouth opening. Its eyes brighten up before a blue flash of light appears, causing the skeleton to move to the right, off the screen. The flash becomes a beam of light, circling around before it reaches the bottom-centre of an outline of the Nintendo logo. It flashes even brighter and the full Nintendo logo appears in light blue. Throughout its screen time, the colour of the Nintendo logo changes to red.
The first is seen on Nintendo-licensed games since 1986.
The second is seen on Super Mario World, Super Mario World 2: Yoshi's Island, The Legend of Zelda: A Link To The Past, Super Scope 6 and the Super Mario Advance games except for the first one.
The third is seen only on King of the Zoo, an European localization of Penguin Wars, for the Game Boy.
The fourth is seen on Super Mario Kart, Super Mario All-Stars (+ Super Mario World), Stunt Race FX, Super Punch-Out!!, Yoshi's Safari, and Super Mario All-Stars 25th Anniversary Edition.
The fifth can be seen on most Nintendo games such as Mario Kart 64.
Thankfully, the sixth only appeared on Eternal Darkness: Sanity's Requiem for the Nintendo GameCube.
None for the first, second, and fourth.
Depending on the variant for the fifth one: Mario Kart 64: Low to medium. The revving up of the cars and the logo spinning may make you uneasy. Goldeneye 007: Minimal, as you might not expect it. Perfect Dark: Low, the dark ambiance might creep you. Animal Crossing: Wild World: None. Donkey Kong 64: Minimal. Donkey Kong saying "Okay" when the logo suddenly appears may startle you. None for the silent/still variant.
Nightmare for the sixth, the smoke, the frightening and jarring appearance of the skeleton growling, the orchestral sting and the creepy announcer will be very unsettling to most. But this is undeniably intentional given the subject matter of the game that follows it.