The easiest op-amp application to understand is the inverting amplifier, which looks like this:
Notice two things. First, we left out drawing the power supplies, V+ and V- in the diagram, even though they're still in the circuit. This is common practice.
Second, see that we're feeding a portion of the output back into the inverting input. This is the heart of the negative feedback loop, which will trades off a reduced gain for clean amplification.
What is the gain of the inverting amplifier? It's easy to calculate:
Gain = -R2/R1
(There's a clear, detailed explanation of exactly why this formula holds true, found in The Art of Electronics.)
Let's take a look at an example circuit:
Let's assume we feed it a sine wave that varies from +0.5V to -0.5V, and let's also assume for now that we're powering the chip with +10V on the V+ rail and -10V on the V- rail. What will the signal out look like?
First, we'll calculate the gain. R2 is 10k, R1 is 1k, so the gain is -10. So our signals will look like this:
This is pretty much as we expect -- the amplifier inverts the signal and multiplies it by ten.
But what happens if we feed the same signal into the same amplifier, but change the power rails so that V+ is just +5V, and V- is now Ground (0V)?
We're going to run into problems:
Here, the output wants to go past the lower power rail of 0V, but gets clipped at about .7V. And it wants to go all the way to 5V, but gets clipped around 4.3V. This clipping happens when the power rails are set too close to the max and min of the output signal, and is something to watch out for.