According to the Daily Puppy, dogs groan as a form of vocalization to help convey the way they are feeling. Because groans are used by dogs to vocalize a lot of different feelings, it is up to the owner to read the situation and determine what each groan means.
One common reason why dogs groan is because something feels good. For example, dogs often groan when they are being petted on the top of their heads. Sometimes this is mistaken for a growl, but it is actually a way for the dog to communicate to the person that the petting feels good.
Dogs also sometimes groan when they have admitted defeat. One example of this is if a dog has been begging for food off the table and the owner does not give him any. Groaning in this scenario is the dog admitting defeat, and usually results in the dog going to sulk.
Due to the fact that dogs are not verbal in a way that humans understand, learning a dog's body language is important for dog owners, according to About.com. Dog owners have to pay attention to not only vocalizations made by dogs, but also their overall body language in general. By putting body language together with various situations, an owner better understands what a dog is trying to communicate at different times.