A vertical slice is a portion of a game which acts as a proof of concept for stakeholders before they agree to fund the rest. It is not the same thing as a prototype in that it is expected to look of final quality and play like the final game. It is like asking to see a piece of the final cake before agreeing to pay for the whole.
Requests for vertical slices are a key sign that your stakeholder (publisher, investor, whomever) has no idea what game making actually involves and is likely harboring many misconceptions about the process. The reason is that what seems like 10% of the content and therefore 10% of the effort is actually more like 70% of the effort.
Those who desire to see vertical slices make the mistake of thinking of game development as a pure production exercise, which of course it is not, and will likely misunderstand what further development will entail. Vertical slices are also simply not achievable in many kinds of game.