In The Grinder, the script context includes a thread number, a process number, and a run number. I believe the thread number identifies a thread within a group of threads spawned by a worker process. The process number identifies a worker processes created by an agent. What is the run number? To put it another way, what actions result in a new run number?
From my experience with JMeter, when you run the load test the Virtual Users execute the same script over and over again for either the specified time period or the specified number of loops.
I would assume that each time the user (thread) executes the script (loop), it is counted as a run.