Conditions are used to make the events more specific. An event could be "a unit is attacked"; without conditions if any unit on the map is hit then that trigger will run. You can use conditions to do things such as specify the type of unit to get attacked before the trigger runs, etc.
If you used to be a StarEdit mapper (like me), just think of the StarEdit Conditions as broken down into Events and Conditions. If you're completely confused, the best way to learn is to open up a good map with triggers that you've played before, and see how everything works. Then read a FAQ for additional information.
Ghan has said he has fixed this. Monovertex please confirm this fix. This was only a problem with people that had signatures in the upper levels like not the special members but the respected members.