When cats blink slowly at you it is a sign of affection referred to as a “cat kiss.”
Slow blinking by a cat (sort of an eyes-almost-closed look, almost trance-like) is a good sign — one that says, “You’re my buddy, and I feel comfortable hanging out with you.” My sister’s cats certainly must like me, as they have never jumped at me, hissed at me or run from me.
I did a little research on this topic, and it turns out that cats tend to do a lot of talking with their eyes; the eyes are part of a feline’s communication system. If you are directly staring a cat down, you’ll only make that cat nervous. They see a stare-down like this as a threat.