In my lab, it's intellectual conversation that soon leads to research.

There's no agenda at first: just a free-form chat about anything that interests us. Often the student has been wondering how some problem could be solved, so I'll suggest approaches, or suggest a more interesting version of the problem. We bat this ball back and forth ... and often we start getting excited about a possible project that we could actually do.

Sometimes this is a project that I'd previously thought about; sometimes it's pretty new. Typically it's somewhere in between - the result of the student's interests making contact with my own preoccupations. (Making contact isn't hard: I've spent the past couple of decades thinking about all kinds of stuff in this field, which is why I get to be an advisor. :-)

Then we'll spend the rest of the meeting working out details of how we'd actually formalize this problem and attack it. That's interesting and educational for both of us, and helps us decide whether the project is worth actually doing. If I'm suggesting techniques that the student doesn't know yet, I'll take the time to explain them and perhaps recommend some reading.

In the next weekly meeting, we might continue on the same topic, or start fresh. But the series of meetings basically forms one long conversation where we tend to circle back to previous ideas. It's pretty fast this way to come up with a project that we're both genuinely interested in. Usually within a month, these meetings are feeling like actual research, in the sense that we're trying to solve a well-defined problem and the student has something to work on between meetings.

p.s. Of course other topics come up during meetings as well. Career advice, coursework, interesting research being done by others here or elsewhere, ... There's a social component too. Some of my students are pretty entertaining; we kid around a lot, or toss around physical objects along with the ideas (yes, there are toys in my office).