Operational amplifiers (OPAMPs) are high performance differential amplifiers in integrated form that can be used in many different ways. A typical OPAMP has a non-inverting input, an inverting input, two dc power pins, one output pin and a few other fine-tuning pins. On the following image you can see a typical diagram of an operational amplifier.
The basic OPAMP operation is simple. If the voltage applied to the inverting input is greater than the voltage applied to the non-inverting input then the output saturates to the negative supply voltage. In addition, if the voltage applied to the non-inverting input is greater than the voltage applied to the inverting input, then the output saturates at positive supply voltage.
This operation mode is limited and doesn’t give us the full idea behind OPAMP operation. The trick to make an OPAMP more useful is to provide negative feedback from the output to the inverting input. In the image below we see an OPAMP with negative feedback working as an inverting amplifier.
In this configuration a part of the output voltage is fed back to the inverting input and thus the gain of the OPAMP can be controlled and output isn’t saturating. The gain of such an amplifier is controlled by the two resistors Rf and Rin. The minus means that the output is inverted relative to input.
By adding more components on the feedback loop, different OPAMP circuits can be made, such voltage regulator circuits, current to voltage converters, oscillators, filters etc.
Beside the negative feedback, a positive feedback can be used. This way the OPAMP is driven toward saturation and works in either +Vs or –Vs output range. Applications of positive feedback is on comparator circuits and oscillators.
Theory
The basic formula that gives the output voltage in respect to input voltages is as follows. This formula tells that the output voltage is a function of the input voltages difference and of the open-loop voltage gain A0.
This expression is for the ideal OPAMP and can get more complex for the real OPAMPs. Some rules apply to better understand the ideal and real OPAMPs.
Read more: Introduction to OPAMPs and Applications
French 
English