Non-stop: a flight from an origin airport which goes directly to the destination airport, without any connections or plane changes. If you want the shortest travel time, fly from your city non-stop. Non-stop flights are usually the most expensive compared to direct or connecting flights.
Direct flight: a flight from your origin city which does not require passengers to change planes. However, while not a non-stop flight, you probably still have to land at a connecting city or airline hub. It just means that the flight number remains the same and you do not have to change aircraft. Often times when you see a flight schedule it may not indicate that there is a connection.
Connecting Service: means your flight from your departure Airport will have at least one connection, and most likely a plane change as well. So that great fare from SLC you found comes at a price!