First of all, the name of the process is voir dire (a term that you can google if you would like to).
In general, the lawyers for both sides of the case will question you – generally in a fairly impersonal and public manner – regarding your knowledge of the case, relationship to any of the parties in the suit, whether you’ve ever been involved in such a suit yourself, whether you have particular biases (or might have) based on your own relationships with persons or businesses like those involved in the suit. If the questions ever get to a “personal” nature, then the courtroom will be generally pretty well cleared for those questions, I believe.
For example, when I was selected for jury duty once, the case was about a person suing his insurance company. When it was established that I didn’t know the plaintiff, didn’t have a relationship with the insurance company (or with anyone who worked there or any of the attorneys, etc.), had never been involved in a dispute with an insurance company, didn’t know the particulars of this case, etc. there was no reason to exclude me, so I was selected.
That was completely impersonal. If you have any personal biases or experiences that bear on the case, then one attorney or the other will want to know that so that they can make a “peremptory challenge” (not challenging you, but your ability to serve as an impartial juror) and they will exclude you “for cause”. (I believe that the attorneys for either side have an unlimited number of challenges “for cause” if they can show cause why a juror should be excluded. They also have a limited number of “peremptory” challenges that they can use “just because” without having to explain a reason.)