Definition of Product Increment:
In each sprint, the development team creates a product increment. This is ultimately a small part of the "final" product. Each Product Increment should be potentially shippable. This means that it does not have to be delivered, but it should be in a deliverable state. The Product Owner decides whether it should be delivered. The Development Team ensures that it is in a deliverable state and thus meets the Definition of Done. Especially teams that have problems with the creation of a Definition of Done can be asked what has to be done so that the product can go out to a customer. In the software context it is usually no problem to create a potentially deliverable product increment. In hardware development it looks a bit different.
Synonyms for Product Increment:
Shippable Increment, Partial Product, Sprint Result, MVP (which is not correct).
Advantages of product increments for project management:
A product increment enables the team to not only present their work result to their stakeholders, but to actually demonstrate it and even let the stakeholders interact with the product themselves. This results in much more meaningful feedback that helps the team learn.