A battery can make electric current flow through a wire. What does the battery contain that can make that happen? Electric current is electrons on the move from one place to another. There are electrons in all atoms, but they usually stay in place. What can make electrons move from one place to another?
It has to do with the fact that different substances pull on electrons with different strengths. There are substances that don't mind picking up some more electrons... ... and substances that would like to get rid of some of their electrons. And this, is something we can use. Here's an atom that wants to get rid of electrons, And an atom that wants to pick up electrons.
If we let the atoms come in contact with each other, a chemical reaction occurs. Electrons move from one of the substances, to the other. What happens if we keep the substances separate, but connect them with a metal wire - a conductor? One of the substances still wants to give away electrons, and the other still wants to pick up electrons. The only way they can do that - is by sending the electrons through the wire.
We have created... A battery. We have used a chemical reaction to create electricity. This area of chemistry is called - electrochemistry. Using chemical reactions to create electricity, is one part of electrochemistry.
The other part does exactly the opposite: here we use electricity to cause chemical reactions. One example is when we have a metal in combination with other substances, and we want to extract the metal in pure form. Metals that are part of chemical compounds have given away electrons - they have formed positive ions. Here are some aluminium ions, for instance. When they have formed a compound, they have given away three electrons each.
If we want aluminium in pure, metallic form, the atoms need to take back three electrons each. The problem is, aluminium is a substance that is unlikely to pick up electrons - the atoms prefer to stay in their ionic form. We have to force electrons on the atoms. How can we do that? By using electricity!
By connecting a wire from a battery, or another source of electric current, we push electrons against the aluminium ions. Then they have no other option but to pick up electrons, and we have forced a chemical reaction to happen - by using electricity. These two types of reaction are the opposites of each other. In the first example, the substances want to react with each other by transferring electrons. We let them get their way - but only if the electrons take a detour through a wire.
In the other example, the ions don't want to react. But we force them to - by pushing electrons towards them. In the first case, electricity is generated by a chemical reaction. In the second case, we make the chemical reaction happen, using electricity. Both are examples of electro-chemistry.