To power up on steemit means to use steem which you've purchased or earned from posting to increase your steem power. If you want to increase your influence on steemit and that's more important to you than getting rewarded immediately and cashing out, then you can set all your payouts to 100%SP, that way, everything you earn will be in SP. Just about every whale on steemit powered up to get to where they are and if you want to become one then you should start powering up today.
To power down on steemit means to liquidate your steem power and receive it in steem. It takes about 13 weeks to complete a power down and you can power down all your steem power. You will receive 1/13 of what you're powering down every week and it will continue like that till you've completed the power down.
I hope this answers your question.