How would the United States Congress be different if three or four parties were to be represented there?

The answer to this would depend in large part on how many
seats were held by the one or two small parties other than the Republicans and the
Democrats.  If those parties held enough seats, the workings of Congress might be
somewhat different.

The basic framework of Congress might
or might not remain as is.  In our current system, the Speaker of the House, for
example, is chosen by a vote of the members that always goes on strict party lines.  If
no party held a majority of the seats, this system might be disrupted.  If all the
parties simply voted for their own leaders, there would be no change as the party with
the most votes would win.  But it might happen
differently.

Let's assume the Democrats (as they are now)
were in the minority in the House.  They might make a deal with one of the smaller
parties in which the smaller party would vote for the Democratic leaders in exchange for
certain concessions.  At this point, we would have a coalition government in the House
where the Democratic Party had to be attentive to the needs of the other
party.

In practice, however, this might not be such a big
change.  Right now, there are factions within parties that have to coexist.  The
freshmen members of the House Republican Caucus are said to be very ideologically
different than the leadership.  This means that the House leadership is already having
to be attentive to the needs of a bloc of members whose views are different from their
own.  When looked at in this way, the presence of smaller parties might not actually
make any difference at all.

All this is to say that it is
hard to know if there would be any really important differences if we were to move to a
situation in which more parties were represented in Congress.