BJT Transistor Amplifiers
The circuit on the right provides a DC bias to the amplifier to place the BJT into the preferred regime of operation. It uses the capacitor as a filter to take only the AC component of the input, and feed that on top of the dc bias.

Doing this allows you to amplify small AC signals effectively without worrying about the regime of operation changing with the input. BJTs are nonlinear devices, but when the input is only changing by a small amount, you can pretend that it is linear.

With the circuit on the left, you are depending on your input already having that bias. It doesn't really matter if this is the entire circuit and you have total control of the input, but when building circuits to process potentially unknown signals you don't have that guarantee.