What You Absolutely Need:
(for any size tank)

1) Tank (Glass or Acrylic)
2) Light (Metal Halide or Florescent)
3) Pumps/Powerheads (for nano tanks <5g, a bubbler may suffice)
4) Source of Calcium and buffer (options include Kalkwasser, Calcium Chloride & Calcium Carbonate, etc.)
5) Test kits for Calcium, Alkalinity, & pH
6) RO filtered or distilled water
7) Salt for reef aquariums
8) Hydrometer or refractometer (Refractometers are preferred)
9) Heater
10) Thermometer

Note: Used equipment is usually fine so long as it's in good shape.  However, do not use tanks, heaters, pumps or any other submerged equipment formally used for freshwater or for a fish-only marine tank.  Some freshwater medications, like ones containing copper, that are safe for freshwater fish (and for fish-only marine tanks) are NOT safe for reef tanks.  Even with extensive cleaning, some of these toxic chemicals can remain absorbed in the silicone sealant of the tank and in other unreachable parts of equipment.  Even if the former owner claims to have never used these medications, it's simply not worth the risk.  If you are buying used, buy used equipment previously used for (and only for) a reef tank (and by "reef tank" we mean any tank housing marine invertebrates).

What You Should Probably Also Have:
(Especially for Tanks 50 gallons or larger)

1) Protein Skimmer
2) Sump/Refugium
3) Live Rock and/or Live Sand
4) Carbon Filtration (if not in a sump, then through a canister filter or some other filter)
5) Test kits for Ammonia, Nitrites, Nitrates

What You Don't Need but Could be Useful:

1) Chiller
2) Ozone(helps with water clarity)
3) Calcium Reactor
4) Luxmeter, pH pen, other electronic testing equipment