A service plan is priced per visit. How the customer is billed for those visits is a separate choice. You make it once for the whole workspace under Settings → Invoices → Billing, and you can set a different mode for one customer on that customer's Invoices tab.
Monthly in advance is the default. On the billing day (the 1st, unless you change it) each customer gets one invoice for the month ahead: the per-visit price multiplied by the number of visits in an average month. A customer who starts part-way through a month is billed the next morning for only the visits left in that month. If a visit they already paid for is skipped, or falls inside a pause, it comes off their next invoice as a credit. An invoice never goes below zero.
Monthly in arrears sends one invoice at the start of the month for the visits completed the month before, with a line for each visit. Per visit sends an invoice when each visit is marked done.
Billing once a month keeps card fees down. Every card charge carries a fixed fee on top of the percentage, so one charge a month costs less in fees than four small ones.
Automatic billing is off until you turn it on. Before you do, the Billing panel shows the next run: every customer who would be billed, the amount, and the reason for anyone who would be left out. The run happens in the morning in your timezone. You can also run a month's billing yourself with one button. A customer is never billed twice for the same month. To undo an invoice, void it.
Each invoice is emailed as a PDF from your mailbox with the customer's link to pay or save a card. With Stripe connected and autopay on, autopay charges the card on file and retries a failed card three times. A customer with no card on file gets the invoice and can pay from it.