Do you know that cells have "lab tables"? Yes have. These are the ribosomes, the small non-membrane bound organelles composed of proteins and ribosomal RNA (rRNA).
Ribosomes consists of large and small subunits. The small subunit has a site for binding mRNA, a P-site for binding peptidyl tRNA and an A-site for binding aminoacyl tRNA. As you can see these sites are used during protein synthesis. Thus, ribosomes serve as the surface or the lab table for the synthesis of proteins.
By the way, the large and small subunits are manufactured separately in the nucleolus and also released separately to the cytoplasm. They remain as separate entities until the synthesis of proteins starts.