Full Answer
The terms sit on a spectrum of how directly the customer chose to share. First-party data is what you observe: the orders someone places, the products they view, the emails they open, the lifetime value that builds over time. They didn't fill anything in to give it to you; it's the exhaust of normal interaction with your store, and because you collected it directly, you own it and can act on it without a broker in the middle.
Zero-party data, a term popularised by Forrester, is what the customer hands over on purpose. A style quiz answer, a declared budget, notification preferences, the reason they're shopping, these are stated intentions rather than inferred behaviour. Because they're explicit, they're unusually reliable and carry clear consent, but you only get them when you ask and the customer chooses to answer, so the volume is smaller.
Both stand in contrast to third-party data, which is aggregated and sold by parties the customer never dealt with, the model privacy regulation and browsers are dismantling. For a WooCommerce store the practical strategy is to capture first-party behaviour properly, server-side so you don't lose a third of it, and to layer in zero-party signals through well-placed questions. Observed plus volunteered, both owned and consented, is the durable foundation that survives the end of third-party tracking.