Proteins are made in a cell primarily in the ribosomes. Ribosomes can either be found floating freely in the cytoplasm or attached to the endoplasmic reticulum, which is referred to as rough ER due to the presence of ribosomes on its surface.
The process of protein synthesis begins with the transcription of DNA into messenger RNA (mRNA) in the nucleus. This mRNA then leaves the nucleus and enters the cytoplasm, where ribosomes read the mRNA sequence to assemble amino acids in the correct order to form a protein. This process is known as translation.
In summary, while the instruction for making proteins is found in the DNA of the nucleus, the actual assembly of proteins takes place in the ribosomes located in the cytoplasm or on the rough endoplasmic reticulum.