To be groggy is to be one of the twelve different varities of sleepy. Grogginess is often characterized by confusion, slow movement and an inability or unwillingness to get out of bed.
Grogginess is usually a feeling that people have after waking up from sleeping, and not after being awake for too long. In addition, grogginess is usually not connected with sleep deprivation, but is usually the result of having had a nice, long period of sleep, and your body or mind just not wanting to go back to the cold, confusing world.
It is interesting to contrast this with actually having to wake up after a too short period of time, when your body and mind are often surreally alert and awake. However, in this cases, they are often more confused and slow to react then a person in a groggy mood.
Grogginess often passes with the addition of some caffeine, a hot shower or just some time. After that, the groggy person is usually comfortably awake.