What is the bare minimum for a functioning prison?

One of the grants you can get right at the beginning is titled "Contract: Basic Detention Center". This grant gives you $40,000, and adds a few items to your checklist. These items are:

• Build a holding cell
• Build a shower
• Build a yard
• Build a kitchen
• Build a canteen
• Hire at least two guards
• Hire at least two chefs

Other than two guards and two chefs, that sounds like the absolute minimum required to incarcerate and detain your first batch of prisoners.

fbueckert's answer is good but does not address all of the prisoner's needs, which should be considered as the real bare minimum for a functioning prison, since fulfilling needs are what prevents riots and escape attempts.

• Bladder,Bowel,Sleep,comfort - a cell/holding cell with a toilet, bench, bed
• Food - canteen, kitchen
• Hygiene - shower
• Clothing - laundry, 3x office (for laundry)
• Exercise,recreation, family - yard, common area with lifting weights, phones, TVs
• Defence,Freedom - guards

So to sum it up, you will need overall

• Holding cell
  • Bench
  • Bed
  • Toilet
• Canteen
• Kitchen
• Chefs
• Shower
• Laundry
• 3x Office
  • Warden
  • Chief of security
  • Foreman
• Common room
  • Telephone
  • TV
• Yard
  • Exercise equipment
• Guards