I paid for my hotel at booking.com. When a hotel runs your card at arrival for any possible future damages , do they charge your card right away ? Or do they keep your card information just in case of anything ?
As a general rule the hotel will generate a "pre-authorization" against the card you provide at check-in. As your room is already paid for, this pre-auth would generally be for a fixed amount of per day (often around $50, but it will depend entirely on the hotel) which is designed to cover things like room service, mini-bar, etc.
If you are using a credit card, this will show as a pre-authorization against your card, and will reduce your overall credit available, but will not show as a charge.
If you are using a debit card, the pre-auth will (temporarily) reduce the funds available in your account.
When you check-out, any charges you have made will be charged to the card against this per-authorization, and the remaining balance will be returned to your credit limit/account - although this may take a few days to occur.