Something that's bleary is dull, tired, or unfocused. Your eyes might be too bleary first thing in the morning to focus on the newspaper — at least before you've had a cup of coffee.
Bleary almost always describes a person's eyes, although you can also use it simply to mean blurred or exhausted. A day spent traveling from Asia to North America can leave you with bleary eyes, and so can a night of crying after the death of your beloved dog. The word bleary comes from a now-obsolete adjective, blear, "watery or rheumy," which may share a root with the German blerre, "having blurred vision."