Buying your own gear is probably the most expensive part of scuba diving. If you only dive a few times a year or even less, there is no need to buy full equipment. If you dive a lot, it is  cheaper to have your own equipment than renting dive gear for longer periods of time. You can also collect your gear over time, or decide to only buy a couple of essential parts, such as a mask, fins and a wetsuit.

Essential gear:

Dive mask and snorkel 
Fins (and boots when open-heel)
Buoyancy control device
Regulator
Exposure protection: wetsuit (or dry suit)
Tank with valve and belt
Weight belt with weights
Dive tables or planners
Logbook
Dive watch
Dive compass

Optional gear:

Dive computer
Knife or tool
Dive light
Underwater camera
Hood
Gloves
Lube suit (in order to get into wetsuit more easily)
Gear bag
Extra clips to attach accessories to BCD
Soft weights
Slate and marker
Defogging solution
Dry bag
Spare parts (o-rings, fin straps, mask straps, etc)

Make sure you are well-informed before you buy any gear, in order to make sure that you end up with equipment that suits your needs and your level of experience. There are many brands and there is a large market, so make sure that you take your time to choose. There is also special gear for women, such as BCD’s and wetsuits, which can be more comfortable than the standard gear for men you find for rent in dive schools.