I want to install a pool table in my basement and I want to know what
kind of flooring I should or should not use, considering the weight of
the table and the  damage pool balls will do should they fly off the
table.

The basement floor is concrete.  

I believe ceramic tile would be fine, if expensive.  Also solid
hardwood would be fine.  High density floating floors might work, but
might dent (from the table or the balls) Low density floating flooring
would not work.  Carpet would be fine, I think.  Suggestions?

Softer flooring will absorb the impact of the falling pool ball. On
concrete and ceramic tiles the pool ball will tend to bounce unless
the ceramic tile breaks or cracks on impact absorbing the momentum of
the pool ball. To a lesser extent they will bounce of wooden floors.
There is a recorded case in Western Australia where a pool ball
falling from a pool table in the public bar area of a hotel bounced
off ceramic tiles and struck a young man a glancing blow to the side
of his head (temple) killing him instantly.
Anything that is not solid for the pool table to stand on is likely to
sink or compress and the table become uneven. Wooden floors, planks,
etc. may transfer vibrations as people move about the room. Smooth
flooring provide less traction for the players feet when they are
leaning at some angle for that difficult shot.
I saw a ballroom on the ground floor of an old house with wooden
floors. At the one end were 8 squares of white marble evenly spaced in
2 parallel rows of 4. Each square was about 6 inches a side and they
came up from beneath the floor to be flush with the floor. I was told
that a full competition billiard table with 8 legs had once stood
there on the 8 squares (actually the top of 8 stone pillars) which
rested on the soil below to avoid the motion as people moved on the
wooden floor and to stop the floor sagging under the weight of the
table. Try carpet.