Symbols refer to: 
IB is the base current IE is the emitter current IC is the collector current
IE = IC + IB
VCE is the voltage between the Collector and the Emitter (generally a positive value for NPN transistors) VBE is the voltage between the Base and the Emitter(generally positive for NPN’s)
β is the DC current gain (β is given in the datasheet of a given transistor as hFE)
IC = βIB
two modes: Cutoff and Saturation
IC = βIB. if IB = 0 then IC = 0. This is called cut-off mode. (in this case, VC is the same as VCC)
IC reaches the maximum value when VC = 0 (the transistor is a dead short to ground). This is called saturation mode.
cut-off and saturation are the two states where the transistor will be dissipating the lowest power
Saturation:
IC = βIB only holds until the transistor is saturated. When a transistor is saturated VCE will be at a value of VCEsat (VCEsat given in the transistor datasheet) In order to saturate the transistor, the minimum IB required is the value that will cause the maximum IC, given the β value of the transistor and the circuit conditions. (with 10% to 15% margin)
Power dissipated = IC x VCEsat
IB = (VBB – VBE) / RB (Ohm’s Law) RB = (VBB – VBE) / IB
When used with a microcontroller: Knowing its logic high value, the required load current and the transistor β value, the value of RB can easily be calculated.
Make sure IB does not exceed the drive current capability of the microcontroller.
Make sure load current does not exceed the maximum collector current of the transistor.
Make sure power dissipation in saturation mode does not exceed the maximum power dissipation of the transistor.
Make sure VCC voltage does not exceed the maximum VCE of the transistor.
Reccomended 20% margins
Transistor h_FE = 1000 = β IC = 5A max IB = 120mA VCBO = 60V VBE = 2.5V @VCE = 3V, IC = 3A VCEsat = 2V @IC = 3A, IB = 12mA 4V @IC = 5A, IB = 20mA VCEO = 60V @IC = 100mA, IB = 0 Pc = 65W
Microcontroller VBB = 3.3V IB = 16mA
Motor VCC = 14V IC = 5A
Calculated
RB = (VBB – VBE) / IB IB 15% = 16mA * (1 - 0.15) = 13.6mA RB = (3.3V – 2.5V) / 13.6mA = 0.8V / 0.0136A RB = 58.8Ω
IC = βIB IC = 1000 * 13.6mA = 13600mA = 13.6A IC = 13.6A
Power dissipated = IC x VCEsat Pd = 13.6A * 10.88V Pd = 147.9W Pc = 65W Pd > Pc !! BAD