Piety is a measure of your standing with your chosen god. You earn Piety by performing actions in accordance with your god's religion. The more you have, the more rewards your god will lavish upon you, in the form of special abilities and gifts. It is measured on a scale from 1-200. If it drops below 1, you are excommunicated and incur penance.
When your Piety reaches certain levels, your god will grant you a new ability. Some of these abilities must be invoked with the a command, and cost MP, satiation, Piety, or some combination thereof. Others are passive, working automatically and (usually) without cost.
You cannot see your exact Piety number in-game, but you can estimate it from the number of stars shown by your god's name on your HUD. Not coincidentally, these stars are the levels of Piety required for you to receive new abilities.
Not all gods give abilities at each level. For example, Okawaru gives the ability to invoke Heroism at *, and then nothing until Finesse at *****. Also, many gods grant powers that improve with your exact Piety (not just your star level), so it is worth cultivating Piety even if you think you've earned every reward on your god's chart.
Along with abilities, many gods will grant random gifts of powerful items to a particularly pious worshiper. Usually, a sufficiently high Piety is all that you need for a gift, but some gods set further conditions for certain gifts.
If you qualify for a gift from your particular god, you have a small chance to receive it every time you gain Piety.
If you receive a gift, you undergo a period of "gift timeout", the length of which varies by god. You cannot receive any more gifts until your gift timeout expires. It will not go down on its own. You can only decrease it with actions that raise your Piety (which instead goes towards reducing the gift timeout).
You can estimate your Piety by looking at the Favour description on your religion screen (command ^).
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Followers of Xom get different descriptions - see his article for details.
Prior to 0.16, gods didn't treat followers' kills the same as the player's, for purposes of piety gain.