Proteins are little molecular machines that do things — virtually everything, in fact — inside the biological cell. There are a very large number of them, and they’re constantly being created and destroyed. Protein synthesis is the production of — funnily enough — proteins, inside the biological cell. There’s a really interesting flow of information — called the ‘central dogma’ of molecular biology — that explains how the information in DNA is read to make an intermediate information-carrier, RNA, which is then directly used by the ribosome to manufacture proteins.
Synthesis is the word that scientists use to describe making a chemical compound (a protein is just a particular kind of chemical compound). Protein synthesis can describe the way that proteins are made within the cell as Jack describes.
A growing branch of chemistry is protein synthesis in a lab setting. Proteins are made up of smaller molecules called amino acids (there are about 20 of these), and the number and order of these gives the protein its very specific properties – it’s a bit like letters making up a word. Chemists will often synthesise proteins (or bits of proteins known as polypeptides) in the lab to try and understand how changing these amino acids changes how the protein behaves. Proteins are now often synthesised in a lab by machine – the amino acids are put in the machine and the scientist then programmes the machine to use particular amino acids in a particular order, in order to make the protein, but previously this would have been done by mixing the amino acids together in a certain order in a flask, and then purifying the product after this was done each time – a very lengthy process!