Quick answer: A cat staring at you is communicating — it may be waiting for food, expressing affection through a slow blink, tracking your movement, or alerting you to something it noticed.
A slow, relaxed blink is a cat's way of expressing trust and affection — sometimes called a cat kiss. If you slow blink back, many cats will reciprocate. A cat that stares calmly and then slow blinks is comfortable and content. Try returning it.
Cats learn quickly that humans respond to their gaze. A cat staring near the food bowl is asking for food. Near the door, it wants to go out. Near you on the couch, it may want attention or to be let into a room. Context usually makes the request obvious.
A wide-eyed, unblinking, tense stare — especially with a rigid body — is different from relaxed gazing. Your cat may have heard something in a wall, spotted an insect, or noticed something that startled it. If your cat stares intensely at the same spot repeatedly, check for pests.
Context and body language together reveal what a stare means. Relaxed, slow-blink stares are affectionate. Tense, unblinking stares mean your cat is alert to something.