DewSays

The cheetah is the world’s fastest land animal, going from 0 to 70 mph (113 km per hour) in just 3 seconds. Even a Porsche 997 can’t accelerate that fast. Nevertheless, the cheetah is the fastest only in terms of absolute speed but they can’t maintain that pace for long. Pronghorn Antelope, the second fastest land animal and many other antelope species can easily outrun a cheetah over a long distance. The cheetah must catch an antelope in the first minute or less of the chase; after that the cheetah has to slow down but the antelope does not (there is sayings that fear have long legs).

Fastest land animals

1) Cheetah 70 mph (113 km per hour)
2) Pronghorn Antelope 61mph (98 km per hour)
3) Wildebeest 50 mph (80 km per hour)
4) Lion 50 mph (80 km per hour)
5) Thomson’s Gazelle 50 mph (80 km per hour)

Fastest water animals

1) Sailfish (Istiophorus platypterus) 68 mph (109 km per hour)
2) Swordfish (Xiphias gladius) 60 mph (97 km per hour)
3) Marlin (Makaira) 50 mph (80 km per hour)
4) Wahoo (Acanthocybium solandri) (47.88 mph 75.6 km per hour)
5) Yellowfin Tuna (Thunnus albacares) leaping 46.35 mph (74. 6 km per hour)

Fastest air animals

Swifts can fly at 106 miles per hour (171 km per hour).

Peregrine falcons are averaging 25-34 mph (40-55 km/h) in traveling flight, reaching speeds up to 69 mph (112 km/h) in direct pursuit of prey, but can dive even faster - at impressive 242 mph (389 km/h) speed and easily catch the swift.

In other words, the short answer to the question “who is the fastest animal in the world?” is: peregrine falcons.

Sources:
http://www.wisegeek.com/what-are-the-fastest-animals-in-the-world.htm
http://wiki.answers.com/Q/What_is_the_fastest_water_animal
http://www.homeworkspot.com/ask/fastestanimals.htm
http://wiki.answers.com/Q/What_is_the_fastest_land_animal
http://www.forbes.com/2004/05/14/cx_mh_0514running.html
http://www.birds.cornell.edu/AllAboutBirds/BirdGuide/Peregrine_Falcon_dtl.html