What kind of question can one ask of the Tarot? Well, almost any -- but I don't find that is good for answering deep philosophical questions about the nature and origin of the universe! It is a practical tool and works best in practical situations.
The Tarot can be used for guidance. It can be used for prediction. It can answer simple questions or shed light on complex situations. It can give indications about time, location, and many other things. It can talk about Love, Money, Jobs, Ideas, you name it -- It all depends on the question and the spread.
As a matter of fact, you really don't need to have a question -- we can just draw some cards and start talking about what they generally mean and see where they may have resonance with something in your life or situation. This is called an open reading, and is best done in person. However, I feel it is the best to have something in mind when we to use the Tarot.
As far as questions go, I feel the best ones to ask are those that can help you gain insight into a situation, or provide advice about possible courses of action in the future. In order to gain the best results from the Tarot, I may suggest rewording or modifying a question slightly.
Many times you will be tempted to ask a Yes/No question. This is something that is very natural to do, as we like to distill all the various facets and unknowns about a situation down to what seems like the simplest and most direct essence, a penultimate issue, one that can be answered simply with a Yes or No. However, the Tarot is most powerful when used to shed light on some of the facets and issues behind such a question. So, if you are tempted to ask a Yes/No question, I recommend that you take a moment to backtrack and see if there is a more interesting and expansive question that you have skipped over. Essay questions are more interesting than True/False or Multiple choice here!
So, I recommend that you take a little time to think about your question, and don't worry if it is open-ended. The answer the Tarot will give you is just the beginning of a possible path for your future.