Two numbers describe a battery for a hobbyist: how hard it pushes, and how long it lasts. The first is voltage. The second is capacity, measured in milliamp-hours.
Voltage
Each battery chemistry has its own voltage per cell. A fresh alkaline AA or AAA is about 1.5 volts. A rechargeable NiMH cell is 1.2 volts. A lithium coin cell is 3 volts. The familiar rectangular battery is 9 volts because it holds six small cells in series.
Connect cells in series, positive to negative, and the voltages add. Four AA cells in a holder give 6 volts.
Capacity
A milliamp-hour (mAh) is one milliamp flowing for one hour. As a rough guide, an alkaline AA holds around 2,000 mAh, a 9 volt alkaline battery around 500 mAh, and a common coin cell around 200 mAh.
To estimate running time, divide capacity by the current your circuit draws. An LED circuit drawing 10 mA from a 500 mAh battery should run for roughly 50 hours. Real life gives you less, because capacity drops at higher currents and the voltage sags as the battery empties.
Which to choose
- 9 volt battery: convenient for small circuits, poor value for anything hungry.
- AA holder: far more capacity for the money, and you pick the voltage by the number of cells.
- Coin cell: tiny circuits that sip current.
Do not mix old and new cells, and never short a battery to "test" it. Use a multimeter.
If you want the fundamentals in one place, with hand-drawn diagrams, start with Getting Started in Electronics, or browse all four books.