Well, an amplifier amplifies your signal ,it means the magnitude and power of the output signal is usually greater than the input signal (Output signal is stronger). For a voltage amplifier we can say Gain=Vout/Vin ,which shows how much the amplifier can make the signal stronger. But it's very important to know , the amplifier is able to amplify the signal only AS LONG AS the amplifier (the BJT) doesn't enter SATURATION area. And VOLTAGE SWING relates to this. Voltage Swing tells you the possible range of the ouput voltage of the amplifier. It means that the output voltage (amplitude of a sine wave,for instance) can NOT be greater or smaller than certain values that are called Swing voltages. When the input voltage (amplitude of input) is too high, the transisitor(s) will enter saturation area and the formula Gain=Vout/Vin won't be valid anymore (there won't be more amplification) so the output voltage never increases more than swing voltages.