You and the customer agree three things before any work starts: the total price, the stages, and what triggers each invoice. That is usually a short payment schedule in the quote or contract. Each invoice then refers back to it, so the customer's accounts payable team can see exactly where the project stands.
There are three common ways to set the triggers.
Milestone billing. Each invoice is tied to a deliverable: designs approved, first fix complete, phase one live. This is easiest for the customer to check, because the milestone either happened or it did not. It works well for agencies, consultancies and fixed-scope jobs.
Percentage of completion. Each invoice bills the share of the work done since the last one. Construction often uses this with a schedule of values: a breakdown of the contract into line items, each with its own value and percent complete. It is accurate on long jobs, but it gives the customer more to question, so back each claim with evidence.
Time-based billing. You invoice on fixed dates, usually monthly, for work done in that period. It is simple to run and suits retainers and long projects without clear milestones. The risk is a gap between what you billed and what the customer thinks was delivered.
Progress invoicing and progress billing are the same thing. In the UK, construction work is usually billed through interim payment applications, and construction contracts there carry their own payment rules. In the US, construction payment rules vary by state and by contract. If you work in construction in either country, check the rules that apply to your contracts with a lawyer.