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The mechanics are a bit complicated, but here are the basics.

By powering up, you increase your 'voting power'. This means that your votes will be worth more , than a person who doesn't have a lot of Steem power.

In return for that, you can't freely transfer your Steem anymore. You still own the coins. But their is a 'cooldown' process required, to make your accessible again. Which we know as powering down.

Is it possible to power down until I have 0 influence?

From my understanding, it's possible. But you will render you account useless. Because most actions on the Steem block-chain require some amount of Steem.