It's easy to treat hitting "publish" on a new Printful product as the end of the process, but that's really just the starting gun — a whole chain of things happens automatically once a real customer places an order, and understanding it helps you troubleshoot when something goes sideways.
When a customer completes checkout on your connected storefront (Etsy, Shopify, or otherwise), that order data gets pushed to Printful's system automatically, usually within seconds to a couple of minutes. Printful then routes the order to whichever of its fulfillment facilities is best positioned to produce it — this depends on the specific product, your customer's shipping location, and current facility capacity, which is part of why two orders for the same product can sometimes ship from different locations.
Production itself typically takes a few business days (Printful publishes estimated fulfillment times per product category, and these are worth actually checking rather than assuming), during which your design file gets printed, cut, sewn, or otherwise applied to the blank product, quality-checked, and packed. This is the stage where a defect, if one occurs, is most likely introduced — which is also why defect claims are handled fairly readily by most established POD platforms, since it's understood to be an occasional production reality rather than something you as the seller controlled.
Once packed, the order ships using whichever carrier is appropriate for the destination and shipping speed selected at checkout, and tracking information syncs back to your connected storefront automatically, which is also what usually triggers the shipping confirmation email your customer receives.
Where sellers sometimes get confused is assuming any of this is instant or that they have granular control over the specific facility, exact production timeline, or carrier for a given order — in practice, POD fulfillment intentionally abstracts most of this away from you precisely so you don't have to manage it. Your actual leverage points are mostly upstream of this process: the file quality you upload, the product and provider options you choose at listing setup, and how you handle customer communication if something in this pipeline goes wrong — not the fulfillment mechanics themselves, which run on the platform's infrastructure rather than anything you do after publishing.