What part of the cell function as factories to produce

The production of a protein is called protein synthesis and this is splits into two parts 'transcription' & 'translation! I'm not sure how much depth you need so I'll be brief:
Transcription:
Occurs in the nucleus
The double helix unzips to leave one strand
The bases are joined by weak hydrogen bonds
The mRNA (messanger RNA) takes this onto the next stage as a copy of the bases has been taken!
Translation:
The copy is taken out into the cytoplasm
In the cytoplasm there are organelles called ribosomes
The ribosome reads the DNA
tRNA is involved in this
They are split into triplets ( three bases) which makes an amino acid & amino acids all code for a protein.
Hopes this helps!! :D