I wanted a 60 amp sub panel in attick to run ac and baseboard heaters (nothing more then 40 amps total). I have 60 amp service wire for this job. The place gave me a 125 amp sub panel box... can I run 60 amp wire to a 125 amp sub box for this? Or do I need a 60 amp box?

You can use the 125 amp panel as long as the breaker feeding it does not exceed 60 amps.

Define a "60" amp wire for me please?

What is the type (insulation), number of conductors of the cable? Are the conductors Copper or Aluminum and what is the AWG (size) of the conductors?