Motorcycle sport
There are a variety of disciplines within the sport of motorcycle racing. The FIM is the international governing (sanctioning) body for motorcycle racing and each nation has its own governing body. Other breakaway/independent bodies exist on both a national and international level.
The motorcycles vary greatly in design for each different discipline. Broadly speaking motorcylce racing can be divided into "on" (ie Tarmac) or "off road" racing (that is not to imply road legality is requiremnt, also several disciplins cross over between the two in variety of extents).
Within each disciplin there is usual an oppertuniy to compete with; Classic machinery (ie non current to various extents), minature machinery (eg Pocketbike Racing ), Sidecars and Quads / ATVs.
Tarmac

Road Racing is the racing of sports bikes (derived from the typical road going machine) on tarmac, either on ciruits or closed public roads (sometimes called true road racing). MotoGP and Superbike are prime examples of circuit racing while events like the Isle of Man TT and the Macau Grand Prix are prime examples of closed public road racing. Within road racing there is a variety of sub diciplines including endurnace racing.
Drag Racing / Sprints
Drag racing and/or sprints is typically where two participants line up at a dragstrip with a signaled starting line. Upon the starting signal, the riders accelerate down a ¼ mile long, two lane, straight paved track where their elapsed time and terminal speed are recorded. The rider to reach the finish line first is the winner. This can occour upon purpose built venues (eg Santa Pod), temporay venues (eg runways or drives of country houses)
Hill Climb
Hill climbing is where a single rider climbs or tries to climb a road going up a hill in the quickest time and/or the furthest up the hill before ceasing forward motion. This events can be on purpose built course but are typcially on closed public roads and private roads. The same concpet is also used off tarmac, usually on steeper hills.
Rally
A road rally is a navigation event on public roads whereby competiors must visit a number of checkpoints in diverse geographical locations while still obeying road traffic laws (not to be confused with car rallys such as WRC).
Is a crossover motorcycle racing discipline between road racing and motocross. The motorcycles are mainly motocross types with road racing tires. The racetrack is mixed between road and dirt courses (in different proportions) and can take place either on closed circuits or temporary venues (such as urban locations).
The riding style on the tarmac section is noticeably different form other forms of tarmac based racing, with different line in corners, sliding the back wheel round the corner and using the leg out straight to corner (as opposed to the noticeable almost touching of the bent knee to the tarmac of road racers).
"Off Road"

The direct equivilant of road racing, off road - ie a number of bikes on a closed circuit racing. The winner is the first rider across the line (genrealy after a given amount of time or laps or combination)
Beachcross'
eg Weston, La Toucet (sp), etc
Snowcross

eg ISDE
Hare and Hounds (article link)
Rally Raid
eg Dakar Rally
Known in the US as "Observed Trials"
Time and Observation
eg The Scott (in Yorkshire)
Indoor Trials
Long Distance Trials
Track Racing
Flat Track racing is done on ¼ mile, ½ mile, and 1 mile dirt ovals, typically held at fairgrounds and speedways. It also includes TT courses which include at least one right turn and a jump. It is the oldest form of American motorcycle racing. Speeds of over 100 miles per hour are common on mile and half mile tracks. The American Motorcyclist Association has been the sanctioning body of Flat Track racing in the US, each year awarding the coveted #1 plate to their Grand National Champion.
Stadiums in Norhtern Europe and Australasia, professioanl sport.
'Grass Track'
Grasstrack is held on oval tracks usually 400m+ (ie longer than speedway, it is called Long Track at world level) using a machine very similar to a speedway bike (still no brakes, but normaly two gears, etc)
Beach Racing
Track racing (most like grass track) on a beach, not to be confused with Beachcross.
Auto Race is a Japanese spercific type of track racing held on an asphalt oval course and seen as a gambling sport.
Hill Climb
As per tarmac but often steeper (eg Pikes Peek in the US) but not nessecarly (eg the classic reincarnations such as Red Marley, in the UK).
This discipline brings together riders from the motocross and track racing fraternity making a classification somewhat irrelevant.
To classify
Land Speed is where a single rider accelerates over a 1 to 3 mile long straight track (usually on dry lake beds) and is timed for top speed through a trap at the end of the run. The rider must exceed the previous top speed record for that class or type of bike for their name to be placed on the record books. See—[1] for an example.