How many ways can seven people sit around a circular table?

In a circular arrangement we first have to fix the position for the first person, which can be performed in only one way (since every position is considered same if no one is already sitting on any of the seats), also, because there are no mark on positions.

Now, we can also assume that remaining persons are to be seated in a line, because there is a fixed starting and ending point i.e. to the left or right of the first person.

Once we have fixed the position for the first person we can now arrange the remaining $(7-1)$ persons in $(7-1)!= 6!$ ways.

It depends on what you mean by "how many ways".

It's not unreasonable to count two seatings around the table which only differ by a rotation as "the same".

On the other hand, if the chairs and the view from the chairs are different, it might make more sense to count those seatings as different.

You can also think of it this way. In a straight line (i.e. seating seven people in seven chairs next to each other), there are clearly $7!$ ways. But when they are joined in a circle, a rotation still counts the same way of seating everyone. You'll notice that there are $7$ possible rotations in this case (since seven chairs). So we partition the result from the straight line into $7$ groups. This is $7!/7 = 6! = (7-1)!$. This idea is also called a circular permutation.