Proteins are made by following the instructions in a gene. A gene is a section of DNA. The DNA stays in the nucleus, so a copy of the gene is made and taken to where proteins are built. This takes place in two stages.
In transcription, the DNA unzips around the gene and one strand is copied to make a molecule called mRNA, in the nucleus. The mRNA then leaves the nucleus and moves to a ribosome in the cytoplasm.
In translation, the ribosome reads the mRNA. Molecules of tRNA carry amino acids to the ribosome, and the amino acids are joined in the order the mRNA says. The chain of amino acids makes the protein.