This guide explains the payment statuses in Event Espresso 3.
This is the default payment status. While Incomplete is set as the default, all attendee information is recorded when they register, but those registrations do not count against the total number of available spaces. That changes when an IPN message is sent from the payment gateway or the payment status for that attendee is updated manually. If you have no credit card processing gateway active and are using the Invoice gateway, the status will change to Pending when the user downloads the invoice.
This means the attendee has not paid, but they do have a space reserved for them in the event. When a user downloads an invoice, their payment status changes from Incomplete to Pending, reserving a space for them but not indicating that they have paid (since that would need to be recorded manually).
This means a payment notification has been received from the payment gateway, or the attendee has been marked as paid manually from the back-end.