Protein synthesis takes place in the cytoplasm of a cell and on the rough endoplasmic reticulum (ER).
Ribosomes bind mRNA and translate it into a protein. They reside in the cytoplasm and bind to the endoplasmic reticulum. In both states they translate mRNA into proteins that are important for cellular processes. Where any particular protein is synthesized depends on the post-translational modifications and where the protein ends up being placed or used. Proteins meant for the ER often have a signal sequence at the very beginning of translation that makes it so ribosomes in the cytoplasm are "captured" as they begin translation. They are stuck to the ER, and the protein they translate is moved into the ER as it is being created.