What is a Charge?
To keep it simple for now (once you get to college physics this will be expanded), charge is electrons piled up, or lack of electrons where you'd expect there to be some. Electrons have negative charge and protons positive charge. A normal atom has the same number of electrons as protons, so no net charge.
On some atoms, the outer few electrons are somewhat "loose". When you have a whole bunch of these atoms all next to each other, like copper atoms in a copper wire, these loose electrons can jump around between adjacent atoms. However, if they jump too far, they leave a positive charge (since a negative one went away) where they left, and a negative charge where they are. This imbalance of charges creates a electric field, which you can think of as a force field that pushes and pulls on electrons. Electrons are pulled toward to positive charges and pushed away by negative ones. This electric field therefore won't let the electrons vacate one location and pile up in another over the space of a few atoms.
A voltage source, like a battery, is something that creates a electric field. If you connect opposite ends of the battery to opposite ends of this copper wire with all the somewhat mobile electrons in it, you can get all the electrons on average to move from the negative voltage end of the wire to the positive voltage end. To keep maintaining the electric field applied to the wire, the battery then pumps the electrons that flow off the + end of the wire back onto the - end of the wire, where they again hop between copper atoms and end up at the + end again.
The mass movement of electrons is called current, which is charges flowing. This is a lot like current in a river is lots of little water molecules flowing. Since the charge of one electron is very very tiny and of little use at our human scale, we use a unit of charge called the Coulomb. However, a Coulomb is just a calibrated pile of charge. In fact, it's about 6.24 x 1018 electron charges worth. Actually it's -6.24 x 1018 electrons since we arbitrarily decided electrons have negative charge.
Again to keep the range of numbers nicer on a human scale, we measure current in Amperes, which is one Coulomb of charge flowing by every second. So if you have 1 Ampere (sometimes "Amp" or the official abbreviation "A") flowing left to right in a wire, then there are actually 6,240,000,000,000,000,000 electrons flowing right to left per second past any one point along that wire.
Now that you have a basic idea what charge and current are, forget about electrons moving with their negative charges. The rest of electronics is all built on Amp and Coulombs. Think of that as the conceptual units of current and charge you'll be using from here on. The fact that these happen to (usually) be based on actual negative charges is irrelevant and just invites confusion.
So now let's go back to that battery that caused current in our wire. A battery is really just a pump for charge. In other words, it can make current. However, there is one more metric that is important to mention here, which is how hard the battery can push. One battery may be able to push harder on charges than another, just like one water pump can make a higher pressure than another. It is this pressure that makes the electric field that makes charges move, which is current. This electric pressure is measured in units of Volts. The more Volts a battery can make, the more current it can cause to flow thru the same resistance. This is just like a higher pressure water pump can make more water flow thru the same size nozzle.
So how can we related voltage, current, and resistance? As you can probably see more voltage (pressure) makes more current (flow), but more resistance (smaller nozzle) makes less flow. To put this mathematically:
current = voltage / resistance
This also gives us a definition of resistance by rearranging this equation:
resistance = voltage / current
The concept of resistance comes up a lot in electronics, so we have a special unit just for measuring it, called the Ohm. In fact, the Ohm is defined as:
Ohm = Volt / Ampere
We have short abbreviations for all three of these quantities since just about all of electronics is based on them. A Volt is abbreviated "V", the Ampere as "A", and the Ohm with the greek letter "Ω".
This equation that relates resistance, voltage, and current is a cornerstone of electronics, and is called Ohm's Law, after the guy who first came up with it.
Let's go back to the first form of Ohm's law I showed, which tells us how much current we get:
In physical quantities: current = voltage / resitance
In common units: Amps = Volts / Ohms, or A = V/Ω
That's a great deal to think about already. Try to wrap your mind around this before going any further. Ask questions here as you need to to understand this. Once you get this, we can go on to all kinds of cool stuff.
Like Ali said, charge is a property (or characteristic or feature) of a particle. The particle could be an atom, or it could just be a part of an atom like an electron or a proton.
Unfortunately, we can't really say much about why particles have this property, or what causes this property to exist. We can only describe some things we observe about this property that we call charge.
Charge comes in two types, which we arbitrarily label as "positive" and "negative".
Positive charges repel each other with a force that we can measure, negative charges repel each other similarly, and opposite charges attract each other.
We find that there are components of atoms called "protons" and "electrons" that are always positively and negatively charged, respectively.
Charge is conserved. That means, in all the experiments we have tried, the difference between the amount of positive and negative charge in a closed system is the same at the end of the experiment as it was at the beginning of the experiment, and we therefore believe this is true of all closed systems in the universe.
Even though we don't know what charge is or where it originally comes from, the description of what it does is enough for us to predict lots of useful things and make lots of useful tools like radios and computers.
Olin's answer is excellent. I would add to it that an analogy might help.
(For the purposes of this analogy let's ignore everything after Newton. General Relativity and the Higgs Boson are interesting but aren't going to help understand charge.)
You probably have an instinctive understanding of mass, specifically gravitational mass. But what is it?
Gravitational mass is a property of matter that causes it to experience a force -- called gravity -- when close to other matter with mass. The individual amount of mass of an atom is very, very small, but we have a lot of them and they can add up to make very large masses indeed.
Though all atoms have a tiny amount of mass, some have much more than others. If you have a bag of hydrogen and a bag of lead with the same number of atoms in each, one will be much more massive than the other.
A numeric description of how a massive object affects another is called the gravitational field. If you imagine a large mass -- the Earth, say -- and imagine a tiny mass -- a ball bearing, say -- magically suspended at a point above the Earth, if you suddenly magically let it move, it would move in a particular direction -- towards the center of the Earth. Imagine the Earth surrounded by a field of tiny arrows, all pointing in the direction that ball bearing would fall. The length of the arrow is how hard the ball bearing would be pulled: very hard near the surface of the earth, hardly at all past the orbit of the Moon. That "field" of arrows is the gravitational field of the Earth.
Charge is very similar to gravitational mass. Like mass, it is a fundamental property of matter. Like mass, it causes two objects to experience a force between them. Like mass, just as some kinds of matter are more massive than others, so too some kinds of matter are more inclined to produce charge than others. Like mass, you can take a large source of charge and imagine a field of arrows around it that tell you in what direction and how strong the force would be on a small charge placed there; that's the electrostatic field.
How then are charge and mass dissimilar? The major differences between charge and mass are:
(1) Mass only comes in one kind, charge comes in two kinds. All mass is attracted to all other mass. Like charges repel, unlike charges attract.
(2) The charge forces are enormously larger than the gravitational forces. Rub a balloon on your hair and stick it to the ceiling. The charge forces in that balloon are enough to overcome the attraction of an object the size of the Earth! (Though admittedly, distances are relevant. Your balloon is thousands of meters away from the center of the Earth and very close to the ceiling.) The force between masses is absurdly small compared to the force between charges.
(3) Charge is extremely easy to move around compared to mass. The movement of charge through a conductor is a significant fraction of the speed of light. (The movement of individual charged particles can be slow; think of it like turning on a faucet connected to a very long hose already full of water. The pressure wave that spurts water out the business end of the hose travels much faster down the hose than the water coming out of the faucet does.)