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There is a concept of Sheafs in topology and Algebraic geometry that i dont know what things we prove using them which are useful in topology or in analysis (eg. Harmonic Analysis). Can any body point out some.

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    @Michael Albanese thanks2012-12-18

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One of the origins of sheaves: Analytic continuation is all about sheaves.

A thorough discussion in sheaf-theoretic terms appears towards the end of Ahlfors's book Complex Analysis.

More generally, look at some book treating Riemann surfaces. I heartily recommend the first chapter of Forster's book.

If you are more ambitious, consult Griffiths and Harris's book Principles of Algebraic Geometry. You will find several topics at the confluence of elliptic regularity theory, sheaf theory and geometry. See the discussion of Hodge's theorem for a striking example.