How does the cash drawer work? Does it combine totals at the end of the day?
Exactly. You choose a cash drawer for every checkout, and then you can run transaction reports for each cash drawer.
Cash drawers are a way of tracking revenue and refunds by location. For example, if you have three adoption desks, each can be given its own "cash drawer". This becomes extremely helpful when you end of day money count is off by $20.00 and you know exactly which desk was responsible for the discrepancy.
How do you track and receive payment for the adoptions?
Payments are received via the checkout process. You can see a checkout flow if you go to your person record and to the Animals tab, click "Add Animal" and choose Adopt. If you follow that flow from there, you'll see how payments would be received. There are a few different reports in the Accounting tab of Quick Reports to track those payments.