For those around during the early days of Nintendo, Excitebike was a staple of nearly every home gamer's collection. Along with other classics like Super Mario Brothers and Tetris, Excitebike was a game that seemed to find itself in nearly every home with a console. The game was incredibly simple. Your character rode a dirt bike either against the clock or a group of other racers. If you completed the course in the specific amount of time you moved on. If not, then well, game over. Players could also create their own tracks; usually placing speed boosts nearly large jumps to watch their rider fly as high as possible.
It was the mid-1980s and entertainment options were limited. If you are under 25, check out Nintendo's page on the game.
Motorcycle-themed games grew from there. The idea of riding a motorcycle presents a certain amount of danger and excitement. Game designers tried to tap into that and as consoles developed so did the games. Road Rash was a popular title for Sega Genesis that came out in 1991. It allowed players to tap into the darker side of motorcycles a little bit. The races were staged as illegal events with police officers interrupting the action. Players could also take a swing and knock their opponents off their bike. If you won the race, you got a cash prize that could be used to purchase a better motorcycle.
The Road Rash series pumped out six different games, ending in 2000. Super Nintendo hosted the GP-1 racing games, but those were never incredibly popular and began to disappear about the same time the Road Rash series ended.
The reality was that racing games began to change. Serious gamers moved more toward the Need for Speed series while more casual race fans moved on to titles like Mario Kart. It appeared that the market for motorcycle-based games had dried up. That is slowly beginning to change, though. Motorcycle games are making a comeback, just in different places.
Mini games: While not their own titles, motorcycle racing appears as smaller games within big titles. The most popular mini game of sorts can be seen in Grand Theft Auto V. I write "of sorts" because, in reality, the mini game aspect here is up to you to create, as you simply have the option of riding a motorcycle during your vehicle-centric missions. There might not be enough content for a full game, but it serves as a nice add on.
Online games: The thrill of motorcycle racing exists in a corner of the gaming industry that you may not have of before. Galabingo has one of the more intriguing titles in Velocity, a motorcycle-themed title where users try to win cash and prizes in a bingo game filled with moto action. You can give it a go here if you're interested.
Mobile games: This is the largest current segment when it comes to motorcycle racing and it has more to do with the devices than the actual titles. Developers have created mobile games where the player tilts the screen of their device to make their character turn. This is a perfect fit for motorcycles as it mimics the feeling of turning the handle bars of a bike, adding to the realism more than say a car racing game would. Appcrawler has an incredibly detailed list if you are looking for such a title.
So while the motorcycle-genre may never reach Excitebike levels again, there are plenty of titles, both big and small, that still capture the fun.