The eye's resolution is generally considered to be 'normal' if two points that are one minute of an arc (one sixtieth of a degree) apart can be seen as separate.
However, the eye has some visual abilities that far out-perform this,
such as the detection of an offset between two lines. This ability is termed Vernier
Acuity (and the eye's fantastic ability is used in the reading of micrometers).
For this task, the best performance recorded (by Denis Levi, who's in the Guinness Book of Records) is just a few seconds of arc (a second of arc is one sixtieth of a minute).
This corresponds to the width of a pencil viewed at a distance of 300m !
Stereoacuity, the ability to discriminate depth by the use of the two eyes, is another example of a hyperacuity.