Coin cell batteries are great for very small, low power projects. These batteries are cheap! Buy them in bulk if you need a lot. They are great for testing LEDs. You'll find these type of batteries hidden inside remote controls, electronic tealight candles and lots of smaller disposable devices.
These batteries are not rechargeable. There are a few, more complex chargeable versions, but the vast majority of coin cells are meant to be thrown away once used.
The chemistries and technologies behind coin cells vary. Some are alkaline, others are lithium. Alkaline coin cell batteries have a nominal voltage of 1.5V. Lithium coin cell batteries, on the other hand, have a nominal voltage of 3V.
Coin cell batteries come in a few different sizes, each with a specially coded name to indicate the size and chemistry. Alkaline coin cells all start with an "L", while lithium coin cells are all prefixed with a "C". The popular CR2032, for example, is a lithium battery (3V nominal voltage) measuring 20mm in diameter and 3.2mm tall. An LR1154 (aka LR44) is an alkaline battery (1.5V) measuring 11mm across and 5.4mm tall.
Coin cells are great for powering an ATtiny or other small microcontroller and LED projects.