A good question! Batteries are an extremely important technology and come in all shapes and sizes They do not need power per se, so much as they provide power. And this has much to do with chemistry and materials science. Although there many types of batteries (so many that I could not do them justice), they all work using a similar premise.

A battery is any device that converts chemical energy into electrical energy. You may have learned in your science classes that matter is made of atoms composed of protons, neutrons, and electrons. We are primarily interested in electrons as it is (moving) electrons that give you current (i.e., electrical energy, plus or minus a few unit conversions). Associated with each element is a characteristic potential. (Strictly speaking, this is a potential difference, with the zero potential taken to be hydrogen by convention. You will learn in future Physics classes that it is the difference in potential energies that really matters.) This potential energy can be thought of as the amount of energy stored in your material that can be used to do something. For example, holding a ball above the ground gives it potential energy (i.e., energy to fall via gravity). Voltage is defined as the amount of energy (potential) per unit charge.

The three major parts of any battery are the anode (-), cathode (+), and electrolyte (liquid or solid). The simplest of batteries that I first learned are more generally called electrochemical cells What happens is an exchange of electrons called a redox reaction (short for reduction and oxidation). Reduction means you gain electrons; oxidation means you lose electrons. This redox reaction induces a current. Let's look at an example

Shown here is an electrochemical cell with zinc (Zn) and copper (Cu), two classic materials to choose. Because zinc loses its electrons easier than copper (i.e., has a higher relative potential), these electrons will flow towards copper when the two materials are connected. This flow of electrons is a current that can power something like a light bulb. Notice that the chunks of zinc and copper are in an electrolyte solution with sulfate. This is what is known as a "salt bridge" and keeps the zinc and copper charge neutral. This system is relatively simple to make- you literally just need some beakers, a chunk of zinc and copper, some wires, and a light bulb.

So now you may ask, why aren't all batteries just zinc and copper? It turns out that different materials make better combinations. Lots of people have worked on finding such new materials and the technology behind batteries has evolved substantially since the development of electrochemical cells.

Perhaps you are most familiar with lithium-ion batteries. These were a pretty exciting technology, as lithium-ion batteries were rechargeable with high energy density, could be discharged/charged many times, and had slow loss of charge when not being used. The difference between a Li-ion battery and electrochemical cell (apart from using solid materials instead of solutions) is that instead of electrons moving about, it is lithium ions Li+. In order for lithium ions to be the carrier for the charge, the anode and cathode need to have lithium in them.

For example, lithium cobalt oxide (LiCoO₂) is used for the cathode. If you look at the crystal structure of LiCoO₂, you'll notice that the lithium (in purple) between layers looks pretty easy to take out and put in. And indeed it is!

Developing better batteries is still very much a huge active area of research. What goes inside a battery can be pretty much anything! As materials scientists, we are always looking for new materials that can store more charge, discharge/charge for many cycles, and last a long time when not used. What are some examples? We have seen examples of metal electrodes and materials that involve more complicated structures. But pretty much any set of materials that can undergo a redox reaction of sorts could act as a battery. In fact, just two days ago I learned that there is research to use bacteria (albeit in fuel cells- similar to batteries but uses a constant stream of fuel instead) as a means of harvesting electrical energy!

Hope this helps!
Best,