I'm just a bit confused about what is actually happening when a wave goes up and down. It's a graph of voltage vs. time, so therefore wouldn't that mean the voltage was going up and down constantly? But why? Also, with capacitors, what do they actually do that results in the wave being smoothed out. I understand that they hold charge, but why does this result in the smoothing out of the wave form? what actually happens inside the circuit?
Yes, the voltage is going "up and down" - this is called "alternating current" and is the power that arrives at your house from generating plants such as dams.
Consider just how that wave you are looking at is generated: in its simplest form, there is a magnet rotating inside a coil of wire. At some point the north and south poles of the magnet are aligned in a direction that produces a maximum positive voltage, then the magnet rotates around and that produces a maximum negative voltage. The waves you see reflect the voltage that is produced in the coil and the wires as the magnet rotates at high speed.
The capacitor is able to smooth out the rise and fall in the following way: when current is flowing and the voltage is rising, the capacitor charges; when the current starts to reverse and the voltage falls, the capacitor discharges back into the circuit - which serves to counteract the falling voltage. This is a highly simplified explanation - a basic book on electronics should answer this and provide diagrams - look for the section on "voltage rectification" and power supplies.
Yes, the voltage is going up and down constantly. We call that "AC" for Alternating Current. That is what is used virtually everywhere on this planet for distributing utility power. The other major type of current is "DC" for Direct Current. That is what comes out of a battery, for example.
As to why AC is used for power distribution, that question has been asked and answered several times here on StackExchange. For example...
- Why are the power transmission/distribution systems AC and not DC?
- Why is there alternating current in my wall socket?
Capacitors basically store a charge, but in the Real World, that basic effect is used for many purposes in electrical and electronic circuits. They are used for "AC coupling" where we want to allow an AC signal (such as audio) to pass through, but we need to block the DC part of the signal. Or, as you suggest, we use them to "smooth out" a signal to make it more uniform, like when making a power supply that rectifies that AC mains power into the DC that most electronic circuits operate on.
There are perhaps a dozen explanations of how a capacitor works and some of them may help answer you question better than others. If you look at the Wikipedia article on Capacitors, and especially the Applications section, you will see a summary of the most typical applications, and hyperlinks to better descriptions of them.