Everything is charged via your onboard account as mentioned by others. You can bring cash, gift cards or put a credit card on your account to pay for these onboard charges - this also includes any charges at
Castaway Cay. That being said, I recently learned that any port adventures scheduled with Disney must be cancelled 3 days prior to sailing otherwise they are considered sold to you. Also, usually any special dining (Remy, Palo) and any alcohol tastings must be cancelled usually 24 hours in advance otherwise those are also considered sold to you.
If you put your onboard account as cash, there is some limit as to how much Disney will allow you to charge before it shuts it off and you have to go to guest services and make a payment to get below that threshold (I thought it was either $250 or $300 but I could be wrong). If you use a credit card then they will charge your card once they get to a certain threshold and keep on going. If you plan to use a gift card, I would either add it to your account early on or go down and pay off charges periodically to keep your charging options open (to me the gift card would work like cash basically).
The only thing that is charged separately from your onboard account would be if you choose to do any OBB - On Board Bookings often called placeholder reservations. These you do directly on the navigator app once on board the ship. You can do 2 of these future placeholders per stateroom and each placeholder has a $250 fully refundable deposit that must be paid at the time of booking via credit card. This is not charged to your onboard account. If you do placeholders, they are good for 2 years from the date you book them during your sailing and if you do not use them, Disney will automatically cancel them at 2 years and refund you to the same card you used to pay the deposit (I know because I have had one or two expire recently and i got the money back on my credit card and it was a nice surprise that month!).
Hope this helps. Enjoy your first cruise!