For your needs it is perfectly enough to set the triggers "Execute" field to "once per market". There is no need to check any additional conditions, the trigger will fire once and will cover the selections that satisfy its conditions.
However, in response to your question, a "run" is an execution of a trigger on at least one selection in the market.
So
Trigger's {trigger name} Number of runs is greater than 0
Means that the trigger has fired at least once in this market and has affected at least one selection from this market.
But
Trigger's {trigger name} Number of runs per selection is greater than 0
Means that this trigger has affected this particular selection that the condition is being checked for. In fact it acts as a filter. If you're checking this condition for all the field, it will let through only the ones that have been affected by the trigger.