This happens when you selected any place other than ship office at the start of the day. Besides talking, which acts like the one after receiving a person in your office, there is a special action here, examine location. The cursor changes into an eye if you move it onto a special hotspot, specific for every location. You can read your examination afterwards. (Note that multiple hotspots are available on some places and it's also possible that no hotspot is available.)
COURT
Here's where the action takes place. First you select the persons to call on court and then confirm your selection. If a person is selected it must have a check beside its name. Note that each person can be called only twice, so decide carefully when you call them.
A spy button (with a picture of an ear) also appears if you have the special spy ability. You can then read which persons the other lawyer will call today. That's useful because you might want to call the same person(s) as he and it would be bad to waste one of two rights to call a person.
When the judge enters the courtroom the interrogation takes place. You can call an objection to the other side's question if you think it's not important or bad for your side. Take care though, because you need high reputation to win an objection if the question is important. If objection gets overruled you lose some reputation (and vice versa if you win).