
One of the more popular title screens of Mario Kart 8, Link.
An Alternate Title Screen is an unlockable feature in a majority of Mario Kart games, featuring a new title screen after certain Grand Prixes- or the whole entire game- is beaten. Multiple or sometimes singular characters are featured, usually alongside their vehicles. In the later games, the screens are also accompanied by the voices of characters besides Mario.
Depending on the game, there can be just one unlockable screen or SEVERAL (usually they are all unlocked at once).
Mario Kart 64[]

Unlocked Mario Kart 64 screen.
The earliest Alternate Title Screen was actually found in the 2nd Mario Kart title, Mario Kart 64. The original pictured Mario, Wario, Bowser, Peach, and Toad racing across an unknown, rather bland-looking track, but the new screen featured Donkey Kong assaulting Yoshi and Luigi with a Green Shell, followed by Mario and Peach on a desert course (possibly Kalimari Desert.)
This screen was unlocked by getting a golden trophy in all cups in all available Engine Classes. The Alternate Title Screen was also permanent.
Mario Kart: Super Circuit[]

Konbanwa! It's evening for the racers.
Mario Kart: Super Circuit, the GBA Mario Kart, had two alternate screens, although they didn't differ much from the original. The primary screen had all 8 racers lined up on a track with clear blue skies. Succeeding further in the game lead to the sky turning to an orange-ish evening, and then into the night sky of Rainbow Road, the floating castle within sight.
Like the previous game, the title screens weren't reversible. The evening one was obtained by beating the Super Circuit Cups with gold, and the night was obtained from beating the whole game.
Mario Kart: Double Dash!![]

I guess the team in the very back is Donkey Kong and Wario in the Parade Kart...?
For the start of the game, we see Mario and Luigi riding on Red Fire, along with 7 other racers, racing on Mario Circuit.
Once you beat the whole game, the title screen showcases Mario and Luigi riding on Red Fire again, along with 9 other racers (some are mixed teams, and one is offscreen at the upper right corner), in what seems to resemble the fan tube section of Rainbow Road.
Like Mario Kart 64, this could be unlocked by beating the whole game with 1st place trophy in each cups, and once again, it's permanent.
Mario Kart DS[]

Mario still has the spotlight... Well, it is his series.
Mario took the spotlight of the first title screen in Mario Kart DS, the first screen to be based on one character. However, the repetitive image of Mario driving his Standard MR changed to a backdrop of all the other 11 racers' character art in tiny slots in the back, but with Mario still taking the spotlight, leaning against his Standard MR.
This one could be unlocked for beating the whole game with a 1st place in each cup, and it's once again permanent and never switches back to the old one.
Mario Kart Wii[]

The game of setting precedents, Mario Kart Wii set the standard for Mario Kart games in the future with Alternate Title Screens featuring different characters and their vehicles in the spotlight: Mario posing with the red Standard Kart M, Luigi standing with his green Standard Kart M, Peach riding her pink Mach Bike, and Bowser roaring next to his Standard Kart L. The previous title screen was Mario and Luigi with Wii Wheels, which never shows up again when the alternate ones are unlocked.
Screens of the characters can alternate any time the game is reset, turned on, or returned to the menu, and they also announce the game's name (or a variant, such as Luigi Kart...) They can be unlocked by beating the game, and removed by deleting the file with that completed game.
Mario Kart 7[]

Following the ways of Mario Kart Wii, Mario Kart 7 also added many alternate title screens for different characters as a variation to the usual Mario, Bowser, and gliding Luigi. Yoshi armed with a Red Shell, Peach gliding in her Birthday Girl, Donkey Kong with a Spiny Shell, Mario underwater/gliding... the onslaught of Alternate Title Screens was truly breathtaking.
These could be unlocked all at once by winning every cup in the game with 1st place, and they cycle through the roster at random. Characters once again announce the game in one variation or another.
Mario Kart 8[]

The Koopalings gang as an Alternate Screen:
Mario Kart 8 has no shortage of alternate title screens, offering a great deal of alternates to the original Mario in a Standard Kart on Anti-gravity... Luigi in his iconic Mach 8, the Koopaling squad, Bowser with a Red Shell of death on the Standard ATV, Rosalina up against her own sapphire Standard ATV, pit stop Toads attending to Mario's broken down Mach 8, Peach and Yoshi for the multiplayer, Baby Rosalina, Baby Mario and an ink-squirted Baby Luigi, a graphically drawn image of Link... and that's not even all of them.
There's different title screens for different play options, such as multiplayer, DLC, and online play, so there's already a few unlocked. However, beating the game unlocks the majority of them, most of them available and rotating randomly for single player Grand Prix.
Mario Kart 8 Deluxe[]
Mario Kart 8 Deluxe features a few reused Alternate Title Screens from the original Mario Kart 8 but a few new ones are also added. For example, there are now photos of Link, Villagers, and Mario about to toss a Shell at some Inklings.
To unlock the Alternate Title Screens, you must win Gold Cups in Mirror mode and 200cc in Grand Prix. If it didn't work, it's because you quit the game before the credits finished. For it to work, redo a cup and wait for the credits to finish.