1 to look at something or someone for a long time without moving your eyes, for example because you are surpeised, angry or bored
stare at
What are you staring at?
stare (at somebody) in disbelief / amazement / horror etc
She stared at me in disbelief
She sat there staring into space (=looking for a long time at nothing)

2 be staring somebody in the face
a) informal if somebody is staring you in the face, it is very clear or easy to notice but you have not noticed it:
The solution was staring me right in the face all along
b) to seem impossible to avoid:
Defeat was staring us in the face

to look at someone for so long that they start to feel uncomfortable and look away