I believe that you can make a trip attack while using *any* weapon, but only some weapons are designated "tripping" weapons.
The difference is this - if you are using a tripping weapon, and you fail your trip opposed check you may drop the weapon rather than be subject to a return trip by your opponent. (they often have a bonus to trip too)
If you are not using a special tripping weapon then it is you yourself who attempts the trip and if you lose, the defender may immediately react and make a Strength check opposed by your Dexterity or Strength check to try to trip you.
( non technical addendum: weapons that are designated tripping weapons normally have hooks or flexible bits)
Does this help