The cellular reproduction process of meiosis was discovered by Oscar Hertwig in 1876.  Mitosis was discovered several years later in 1882 by Walther Flemming.
Hertwig studied sea urchins, and noticed that each egg contained one nucleus prior to fertilization and two nuclei after.  This discovery proved that one spermatozoon could fertilize an egg, and therefore proved the process of meiosis.  Hermann Fol continued Hertwig's research by testing the effects of injecting several spermatozoa into an egg, and found that the process did not work with more than one spermatozoon.
Flemming began his research of cell division starting in 1868.  The study of cells was an increasingly popular topic in this time period.  By 1873, Schneider had already begun to describe the steps of cell division.  Flemming furthered this description in 1874 and 1875 as he explained the steps in more detail.  He also argued with Schneider's findings that the nucleus separated into rod-like structures by suggesting that the nucleus actually separated into threads that in turn separated.  Flemming concluded that cells replicate through cell division, to be more specific mitosis.
Matthew Meselson and Franklin Stahl are credited with the discovery of DNA replication.  Watson and Crick acknowledged that the structure of DNA did indicate that there is some form of replicating process.  However, there was not a lot of research done on this aspect of DNA until after Watson and Crick.  People considered all possible methods of determining the replication process of DNA, but none were successful until Meselson and Stahl.  Meselson and Stahl introduced a heavy isotope into some DNA and traced its distribution.  Through this experiment, Meselson and Stahl were able to prove that DNA reproduces semi-conservatively.