Bob-ombs first come into play in the Mario Kart series in Mario Kart 64. They were not used as an Item, but are sprawled out across the Track in multiplayer only. Also, in the Balloon Battle mode, if a player died in at least 3-player, they were transformed into Bob-omb Cars that can ram into surviving players.
In Mario Kart: Super Circuit, these bombs just spectate the award ceremony.
They come back in Mario Kart: Double Dash!! as Wario and Waluigi's special item. Petey Piranha and King Boo also can possess this item because they have the ability to get any special. Since then, a Bob-omb has been an item in the succeeding games, open to every character.
In the game Mario Kart 8 the Bob-omb will now walk around the track whilst ticking away. Oddly, the sound is the sound of it winding up instead of winding down. Previously, they only made ticking sounds while they sat on the track waiting to explode.
The Bob-omb is an attack Item. It can be thrown ahead of the user's kart to try to knock over the opponent up ahead. If no one is near the Bob-omb, it will take about four seconds to detonate automatically. However, if a racer is near one, or if it thrown and lands directly on a racer, it will blow up right away. The blast radius is about 10 feet, and throws any kart up into the air (similar to a Blue Shell.) However, if a racer drives into the explosion, they will simply spin out.
The Bob-omb can also be dragged and dropped behind the kart. Using it as a shield against other Items such as a Green Shell or Red Shell is a bad idea (really bad idea) because it will blow it up right behind you. One strategy is that when you get a Bob-omb early in the race (when players are usually close together), drop it right behind you so multiple racers will be enveloped in the explosion.
The Bob-omb can be used as a defense against Green Shells and Red Shells, but it is rare to have a Bob-omb in first place. If you do, simply drop it behind you right when the shell is about to come down on you, and it will destroy the Item (extremely helpful technique!). However, the Bob-omb's explosion effects the mega mushrooms, resulting in a giant flipping kart.
In Balloon Battle, a good strategy is throwing a Bob-omb into an area with multiple players as soon as possible, causing most of them to lose balloons.