I'll do my best to give my take on it. We always hear the word "Voltage," like a 9V battery, but most of us take it for granted. What does it really mean? Below is a very long answer that I encourage you to read, but a quick answer is:
Voltage is the change in available, or potential energy between two points of a circut. It is similar to the water pressure in a hose. An even closer analogy would be if you were able to make yourself very tiny and go inside a water spigot, just before where the hose is actually attached and walk around. Depending on how much pressure was in the system, it would be very difficult to walk upstream against the pressure. The higher the pressure, the harder it is to walk up it as you need to expend more energy. The amount of energy you put in is also available to take out. This pressure is similar to voltage. Notice that just because you felt very high water pressure, it does not mean the water that comes out of the hose will have a lot of force, since what is attached could be a very small garden hose on mist spray. Likewise, just having high voltage does not mean a lot of power is available. What is attached to the battery or power source plays a huge part too (think of the hose nozzle).
For true science geeks, please read further ...
Here is a more detailed explanation. You first need to understand the concept of "Charge". Charge is defined as the number of electrons ("-") on a mass. A long time ago they defined this number as a Columb, which is about 6.24×10^18 electrons. Don't ask me to define what an electron is, since nobody knows. They exist on all atoms, but they could be a light wave, particle, or something in between. We just don't know, but can measure them with great accuracy and even know their mass from indirect experiments. It is similar to how we can measure and sense gravity, but have absolutly no clue how 2 large objects can sense each other over space and communicate (mind boggling really). The same concept holds for charges, but they repel each other for some very mysterious reason.
You will also hear the term positive charge. Positive charge does not exist. It just means a lack of negative charge. When we put a "positive" charge near a "negative" charge, they always attract each other, even in a vacuum. Again, nobody knows how they do it with no medium to carry the information. Atoms have protons that are called positively charged by definition, but we could have assigned them "-" and electrons a "+" ... it all goes back to Benjamin Franklin who set some of the rules in place and we still follow them today. The +/- stuff is very arbitrary.
So now that you know about charges and the forces that happen between them, we can move on to another important concept called an Electric Field. An electric field defines a continuous set of points in space where if you put a tiny positive charge (if one existed) on any point where charges exists, you would get a certain amount of force. The force is related to the inverse square of the distance, but that is not important. Now, as you can imagine, it must take energy to move this tiny test charge, since it will be either pulled or pushed. This energy is called the Electric Field Potential.
We are now ready to understand Voltage. Voltage is the Electric Potential Energy per unit of charge. A unit of charge is a Columb, as I mentioned before, that has an arbitrary set number of electons. Conceptually, a single Volt is the amount of energy you would expend to move these electrons in space when inside an electric field. The higher the electric field, the great the voltage for this same unit charge test mass being pushed around (9V requires 9 times the energy). The units are Joules/Columb (Joules are a measure of energy).
Voltage does not tell you how long it will last or how much power can be delivered, since that relates to the storage capacity of the battery and what is connected to it. It could provide electrons for 1 second, or years if it was very big. 9 volts relates to the amount of energy that is available for a given amount of charge being moved. This energy can be converted to work, like powering a computer, or running an electric motor. The speed at which the energy can be delivered is another topic and relates to Power. We usually think about batteries in terms of power it can provide, since a 9 Volt battery might deliver electrons much slower than a 1 Volt battery depending on the actual circuit.