Electricity is all about electrons - some of the tiniest bits of matter. Electrons are electrically charged, and in fact are the source of electrical charges. They spin around all atoms. For each electron in the atom, there is one much larger particle called a proton. The proton carries a charge opposite that of the electron. The two charges cancel one another. Without the charge, there is no electrical attraction. Benjamin Franklin made up names for the charges. He called one the negative charge and one the positive charge. The electron has a negative charge and the proton has a positive charge.
Now imagine that you walk up to an atom and pluck out one of it's electrons. The atom now has a positive charge, since it has fewer negatively charged electrons than positively charged protons. If another electron passes by, it will be attracted to the atom and will join it, making it electrically balanced, or neutral again. You can also make an atom negatively charged, by pushing another electron into it.
The overall effect of many extra electrons is an electrical charge. Electrons can then move between atoms. An electron will hit one atom, and the atom will push out one of it's other electrons. When there is a negative charge and a positive charge connected together with a wire or other conductor, the charges will try to neutralize, and all of the extra electrons will flow from the negative charge to the positive charge. This is called current flow. You can see it in lightning, or the spark from your finger tip from static electricity.
So how do we get electricity to work for us? If we put something that is affected by electrons, like a lightbulb, in the way of the electrons, they will push right through it, causing it to light up.