The first rocket ever sent to space probably carried bacteria or some other accidental passenger. But the first animals ever intentionally sent into space were fruit flies launched aboard a V2 rocket in 1947. US scientists were studying the effects of radiation at high altitude.
A rhesus monkey called Albert 1 became the first monkey launched into space on June 11, 1948; also on board a US-launched V2 rocket.
These were just suborbital flights, though. The first animal to actually go into orbit was the dog Laika, launched on board the Soviet Sputnik 2 spacecraft on November 3, 1957. Unfortunately, Laika died during the flight.
At least 10 more dogs were launched into space and on sub-orbital flights by the Soviets until April 12, 1961, when Yuri Gagarin became the first human in space.
Since those first historic launches, many monkeys, chimpanzees, rats, mice, frogs, spiders, cats and even a tortoise were launched into space.