Operational Amplifiers (Op Amps) come in different forms and can be used for different functions. The most basic parts of the Op amp are the inverting input usually identified in schematics with a (-) sign, the non-inverting input usually identified with a (+) sign and the output voltage.
The output of the Op amp is connected back (feedback) to one of the inputs in combination with resistors and/or capacitors to perform the desired "Amplifying" function. The combination of the resistors and/or capacitors used in the feedback along with any connected to the inputs determine what is called the "gain" which you can think of as a sort of multiplier. A voltage signal sent to the input (Vin) has its value that is multiplied by the gain (G). The product voltage is sent to the output as the output voltage (Vout). The only limit to the output voltage is that it can not exceed the voltage used to power the op amp itself.
Vout = Vin x G
So for example, if a voltage signal of .01 Volts is sent into the op amp that has a gain of 500. The output voltage would be 5 Volts. If an input voltage of 1 Volt was sent into the same circuit you would not get 500 V out if the voltage powering the op amp is only 15 V. More than likely you will get something just under 15 V. The output can not exceed the power supply voltage of the op amp.
The function can vary greatly which is what makes Op amps so useful. Some functions are as simple as voice/sound amplifiers in which case you would need to use Audio Op amps. Others are more technical such as differential amplifiers that will amplify the differnce in two signals or summation amps that take multiple inputs to create one summation Vout like a sound mixer.
Even though the study of Op amps can be a college course (or two) in itself. They are also simple enough to use for the everyday electronic hobbyist. You may want to start at the hobby level in your search for more info on Op amps before you try to tackle the theory behind them.
The last source listed is a nice video that leads you into some of the technical applications of Op amps.