To add to pohnpei's response, the fastest reproducing organisms are unicellular ones such as bacteria. They reproduce asexually; that is, they simply divide by mitosis and create two one-celled organism. Each time this happens, the population doubles 1 to 2 to 4 to 8 to 16...and so on. Along with their simplicity, this is a reason bacteria are used for DNA technology. When a new gene is introduced to the bacteria (for instance, for insulin), the new protein is produced quickly, as each new generation of the bacteria also is capable of producing insulin from the engineered gene.
I believe that what this question is talking about is how quickly animals reproduce. Do they have many offspring in a given period of time, or do they only have a few?
Organisms that have use a fast reproductive strategy are generally smaller organisms that have short lives. You can see this in fruit flies, for example, or in rabbits (much bigger and longer-lived than fruit flies, but small compared to other mammals).
Compare that to organisms like whales and elephants. Big animals that live for a long time do not reach sexual maturity until the are older and then have many fewer offspring than smaller organisms.
The terms "slow" and "fast" are relative, so it depends on exactly how slow and fast we're talking about. Organisms that are capable of reproducing at a rapid rate are those that are capable of asexual reproduction like bacteria who merely split in half by fission. Organisms that reproduce more slowly would be mammals who have a gestation period of a couple weeks as in opossums, or months as in humans, or up to two years as in elephants. Mammals sexually reproduce and once fertilization takes place that cell (called a zygote) undergoes mitosis. One round of mitosis takes approximately 27 hours to complete. This creates a bundle of cells we call an embryo which undergoes this process over and over again creating a fetus whose cells will continue to undergo mitosis until it is a fully developed multi-cellular organism ready to be born.