A gasket or oil seal.
When the engine is off all the oil runs to the bottom and sits in the sump, so it can't leak out. When the engine is running the oil is pumped all around the engine to lubricate things, hence it can leak out.
What you need to do is get in there with a fag and clean the oil of the engine, then run it for a few minutes and see where the oil is starting to ooze from.
Many places where oil can leak.......if you can't easily spot it, have your regular mechanic take a look. Some are really hard to trace and they'll put a tracer dye in the oil and look with a ultra-vilet lamp to see where it comes out. Simple places it may leak include: valve/cam cover gasket, oil filter loose or bad gasket, oil pan gasket, loose oil drain plug, pressure-sender, oil cooler lines to radiator. None of those very expensive to fix. More difficult: leaking crankshaft seals. And many more !!
Of course, if it's leaking, check your oil level more frequently so you aren't running with too little.
You might have a bad gasket. Oil usually leaks when the engine runs.
Bad oil seal/gasket somewhere - figure out where the oil starts (drips down remember) and that will likely be the part that is leaking.
Check the oil filter.
Nine times out of ten: loose filter. That is a high pressure point.
Who is the "thumbs downer"?
You know where to put that thumb?
What might be the problem if oil leaks on the ground when my car is on? It slows down and stops when the car is off.