There are many organizations around the globe that try to stop animal testing.

The organization that is best-known is PETA (People for the Ethical Treatment of Animals). It is the largest animal rights organization in the world, with more than 2 million members and supporters. PETA focuses its attention on the four areas in which the largest numbers of animals suffer the most intensely for the longest periods of time: on factory farms, in the clothing trade, in laboratories, and in the entertainment industry. They also work on a variety of other issues, including the cruel killing of beavers, birds, and other “pests” as well as cruelty to domesticated animals.
PETA operates under the simple principle that animals are not ours to eat, wear, experiment on, or use for entertainment.

Another organization is ALF (Animal Liberation Front), but this is a resistance that takes direct (illegal) action with the aim of animal liberation. These activists remove animals from laboratories and farms, destroy facilities, and arrange safe houses and veterinary care. The organization has no leader and consists of small groups ore sometimes just one person, so the group is difficult to stop.

In Belgium you have the ADC (Anti Dierenproeven Coalitie) or AAC (Anti-Animal Coalition). They only focus on the animals used for animal testing. For example they stopped the tobacco laboratory of Philip Morris in Haasrode. Here the animals had to inhale cigarette smoke for 6 hours a day, 90 days long and were killed afterwards to examine the longs and other intestines.

From:
http://www.peta.org
http://www.animalliberationfront.com/
http://www.stopdierproeven.org