When you use a dishwasher to clean dishes, you say that you "ran" the dishwasher. For example:
A: Are the dishes in the dishwasher clean?
B: Yeah, I ran it this morning.
Some other machines that you "run" include:
Run the washing machine
Run the coffee maker
Usually the machines that you "run" should have a process with a clear starting and ending point like the ones above. But some people also use "run" for other machines that don't have a clearly defined process, like:
Run the vacuum
Run the fan
People also "run" computer programs. You can think of this as having a similar meaning to "running" a dishwasher: you start the program, and then it does some work and stops when it's finished.
Sometimes you can also use "run" to talk about the thing that a machine processes:
The towels smelled bad, so I ran them a few times.