Smiling is almost certainly a sign that she likes you.
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A girl staring at you and smiling could indicate that she is interested in you or finds you attractive. It could also be a friendly gesture to acknowledge you. If you are interested, you could try smiling back or approaching her to see if there is a connection.
smile back means she likes you if you like her hit on her or she could just be be creepy
If a girl stares at you and looks down and keeps smiling she may be attracted to you, or may think you look funny. It is hard to say for sure without hearing her side of it.
Girls smile at you for different reasons: maybe they find you attractive or maybe your appearance puts people in a happy mood.