1. An intense stare fueled by an equally intense emotion, e.g. love, hate, lust, concern.
2. An uncharacteristically long and concentrated stare; from the eponymous "Care Bear Stare," a special ability of Care Bears that allows them to shoot light from their belly symbols.
Jack: Hey man, who's that girl over there giving you the Care Bear Stare?
Ray: Oh, shit! That's the girl from the other night. I hit it then split it, and she's been blowing up my phone ever since. Let's dip before she comes over here.