Batteries are generally composed of two electrodes (the anode and cathode), electrolyte, and a semi-permeable barrier. Traditionally, these batteries are pictured as a pair of discrete electrodes in a bath of acid. However, battery cells are almost always designed with the electrodes as thin, flexible sheets. They are then rolled together with a separator in between, and then submerged in a comparatively small about of electrolyte.
Batteries work by transferring particles from one electrode to the other, while causing electrons to flow from one external battery terminal to the other. The anode undergoes oxidation, which generates electrons (electricity). Meanwhile, the cathode undergoes a reduction reaction, which takes in electrons. On the outside of the battery, these electrons attempt to flow from the negative terminal to the positive, creating an energy flow. Placing a load in the circuit, between the negative and positive, forces the electrons to pass through the load item, and perform some kind of work. Meanwhile, the internal electrodes continue to react as long as electrons can flow from one electrode to the other, and the required materials are still present.
Some batteries are rechargeable, which means that this process can be easily reversed. By forcing electrons to pass through the battery in the opposite direction, the particles which moved from the anode to the cathode are pushed back to the cathode. Once complete, the external power source can be disconnected, and the anode will again try to transfer particles to the cathode, when the external terminals are again connected.
These are some common sizes used for flashlights and electronic devices. They are described by the chemistry used in the battery. Single-use batteries are usually described as alkaline while rechargeable batteries may be Nickel-Cadmium (NiCad), Nickel Metal Hydride (NiMH), or Lithium Ion (Li-ion). 
Large batteries are usually of the rechargeable lead-acid type. Applications include starting batteries for automobiles and deep cycle batteries for leisure use. Starting batteries are usually 12 volt batteries containing six lead-acid cells. Large vehicles, such as trucks and buses may have two 12 volt batteries connected in series to give 24 volts.
Modern batteries have an assortment of problems--from wearing out to burring up (literally). They are usually the heaviest components in mobile electronics, and they are also usually the first parts to wear out. The goal is to create batteries which can store a great amount of chemical energy, do not quickly wear out, and are safe to use. Lithium anodes offer this option, since Lithium is such a small atom, which is also in plentiful supply. However, unlike its predecessor materials, Lithium metal forms dendrites (fibers protruding from the surface) as it gives up and regains atoms. As these dendrites grow, they eventually can connect to the cathode, causing a short-circuit. When this happens, the battery rapidly discharges and heats up, quite possibly to the point of combustion. The work-around for this problem which is currently being used is placing lithium ions in graphite as a "framework". This is two-thirds less efficient than a solid lithium metal anode, but still is better than that the alternatives. This "dendrite problem" has been the cause of a sort of development race among battery designers--whoever finds an efficient way to stop the formation of dendrites stands to make a lot of money.