What are the functions of the nucleus, ER, Golgi, and lysosome?

Nucleus — stores DNA, and coordinates cellular activity including growth, protein synthesis, and reproduction. Is surrounded by the nuclear membrane, which keeps DNA in the nucleus and protects it from materials in the cytoplasm.

Rough Endoplasmic Reticulum — studded with ribosomes and produces proteins.

Golgi Apparatus — processes and packages substances produced in the rough endoplasmic reticulum and secretes these substances to other parts of the cell or to the cell surface.

Lysosome — principal site of digestion. Used to break down and recycle cell parts, to undergo apoptosis (programmed cell death). Generally not found in plant cells.

These organelles are enclosed by internal membranes within the larger cell membrane, which helps minimizing competing interactions and increases surface area where reactions can occur.