A ridiculous amount of scientific work from the 50s and 60s, culminating in a handful of Nobels, can be summed up in the following animated diagram. Recall that, in the nucleus, RNA polymerase transcribes the information in DNA to RNA. In the case of a gene product, the RNA is messenger RNA, or mRNA. That mRNA gets processed (not shown here--more later) in the nucleus, and then exported through nuclear pores into the cytoplasm, where little molecular robots called ribosomes translate the mRNA into protein.  

Figure. The protein assembly line. DNA is transcribed into mRNA in the nucleus, and translated to protein in the cytosol. It is instructive to compare this animation with the Central Dogma.