Although you may feel a strong connection and get along extremely well with your new boyfriend or girlfriend, you still need some concrete information. This information helps you nurture the connection with real facts and helps you get to know your new partner better, which is what you need for a healthy relationship. Asking the right questions helps you know what you're getting into. When you are able to see your partner's perspective and how they handle things, you can be more confident about the future of your relationship.
Hypothetical, harmless questions are fun because they show parts of your partner's personality that would not usually present themselves. For example, you can ask, "Would you rather be a penguin or a turtle?" then ask why. These are fun because they don't ultimately mean anything, they stimulate conversation and they give you interesting insights into what kind of person you're dating.
"What if?" questions are also fun. You could ask, "What if you won the lottery? What would you do with the money?" These questions stimulate conversation and teaches you what kind of person you're dating. "I would spend it all on the party of the century" reveals a very different personality than "I would put it in my retirement fund and never touch it until I turn 65."
Questions about someone's past are best asked when the topic presents itself naturally. For example, a person writing about long-distance relationships could use that opportunity to ask his nearby girlfriend if she's ever been in a long-distance relationship. This lets you learn more about the person you're dating without making her feel like she's being grilled.
People's plans for their futures also reveal a great deal about them. This is another situation in which you should let topics rise naturally. If, for example, you're reading an article about small-business owners, you could ask your boyfriend if he would ever want to own his own business. If you see a family with kids, you could ask your girlfriend if she would ever want kids. By easing into these questions through the things around you, you keep it light, casual and fun rather than making your partner feel like he's on the spot all the time.