I will be traveling to Russia this summer and staying for about 2 weeks before traveling to Armenia. My round trip flight is for Moscow, so I would need to fly back to Moscow to go back home. Would I need a multiple entry visa in this case?
Speaking generally, yes, your visa will need to be valid for at least two entries. (Most countries will issue dual-entry visas as well as multiple-entry visas; an image search indicates that this seems to be true for Russia.)
If you can fly back through Moscow without needing to pass through immigration control, however, you might be able to travel with just a single-entry visa. Having a round-trip ticket to Moscow implies that this might be difficult, however, since it implies that you are flying from Armenia to Moscow on a different ticket. This typically means that you need to retrieve and re-check your luggage, which typically requires passing through immigration control.
To help ensure that your visa meets your needs, include the Armenia trip in the itinerary you submit with your visa application, and indicate clearly that you will be entering Russia twice.