Proteins, in eukaryotic cells (plants, animals, fungi) are made during protein synthesis, in which a gene encoding a specific protein's amino acid sequence is first transcribed in the nucleus, from DNA into messenger RNA (mRNA), then the mRNA is translated into the amino acid sequence at a ribosome.
The ribosome may be free-floating in the cytoplasm, or attached to the rough endoplasmic reticulum (rE.R.), depending on where the protein will end up.
If synthesized at the rE.R. then the protein will be packaged into an E.R. vesicle, which buds off and travels to the Golgi apparatus, where it attaches and the protein is further modified at the Golgi. This may include further 3-dimensional folding, adding of other moieties such as sugars (to make glyco-proteins), etc.
After post-translational modifications are complete, the modified protein can then be exported in a Golgi vesicle, budding off to travel to, for example, for export via exocytosis out of the cell. This happens for protein hormone secretions, enzyme secretion, and many other protein destinies.
