A cornucopia of doubts on ballasts?
In lighting, The letter "T" represents the shape of the bulb, which is tubular. The number represents the diameter of the bulb in eighths of an inch. I/E: A T5 is a tubular bulb that is 5/8" diameter. Lengths of these different bulbs (T5, T6, T8, T9, T10, T12) can vary because the wider the bulb, the longer the bulb can be manufactured. There are also some "D" and "U" shaped bulbs. This is why different fixtures require certain bulb types.

Fluorescent tube bulbs have low pressure gas in them (argon) that require ballasts. When power is introduced, the bulb warms up the gas. The ballast then kicks up the voltage in order for the light to start burning brightly. Old fixtures required a ballast and a separate starter to get the bulb brighter, but newer ballasts have the starter built in. Once the bulb is fully lit, the newer ballasts will reduce the line voltage to a normal operating level. 

Fluorescent tube fixtures usually require a specific ballast and tube type. Some fixtures can be upgraded to use a different bulb type but the ballast and the sockets would have to be changed. I have mixed different brands of ballasts with different fixture brands. Bulbs can be any brand as long as the voltage and wattage fall within the parameters the fixture calls for. If you want to save energy and modernize your fixture you can eliminate the need for a ballast by switching to LED lights and bypass the ballast.