An Inquiry is created when you have enabled the "License" download type and the "Contact Me" checkbox under Download > Product Pricing in a Collection within your Fotomoto Dashboard.
In this scenario, when a customer clicks the Download link via your Fotomoto Buy Buttons, they will be shown a form where they can enter information about who they are and for what they'd like to license your photograph for. This information is sent to you, the artist, and is called an Inquiry. At that point, the artist can communicate back and forth to negotiate.
There is currently no way for an inquiry to be deleted. This a feature, not an issue, because it's more likely that an artist would want to keep a history of the negotiation with the client.