The process of transcription primarily takes place in the nucleus of the cell, where the genetic material, DNA, is located. Specifically, within the nucleus, the nucleolus is a prominent structure involved in the synthesis of ribosomal RNA (rRNA), which is a critical part of ribosome production.

However, it is important to note the distinction that while general transcription of mRNA occurs in the nucleoplasm (the fluid surrounding the nucleolus), the transcription of rRNA genes specifically happens within the nucleolus. The nucleolus is the site where the ribosomal RNA genes are transcribed, then processed, and assembled into ribosomal subunits.

The other options are involved in different cellular functions:
- The mitochondrion has its own DNA and can perform transcription of mitochondrial genes, but this is separate from nuclear transcription.
- The rough endoplasmic reticulum is mainly involved in protein synthesis through its ribosomes.
- The Golgi apparatus functions in the modification and packaging of proteins, not in transcription.

Therefore, the correct answer is the nucleolus, because it plays a central role in the transcription of rRNA within the nuclear structure.

Reference:
Molecular Biology of the Cell, 6th Edition, Alberts et al., Chapter 6: Inside the Cell / Page 250-255