Algebraic Geometry for non-mathematician

MR1075991 Abhyankar, Shreeram S. Algebraic geometry for scientists and engineers. AMS Providence, RI, 1990. It is really for scientists and engineers.

Another excellent choice is D. Cox J. Little and D. O'Shea, Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics. Springer, New York, 2007.

Griffiths Harris is far too difficult, especially for "non-mathematician". Shafarevich is OK for a beginner but beginner-mathematician.

EDIT. B. Hassett, Introduction to AG, Cambridge 2007 is a very short book in the spirit of Cox, Little and O'Shea, aimed at undergraduates. Grobner bases are emphasized.

Another excellent book, in a very different style (classical geometry) is J. Harris, AG. A first course, Springer, 1992, 1995.

You might be better off studying an analytic approach to algebraic geometry first. The classical reference for this approach to algebraic geometry is Griffith and Harris' Principles of Algebraic Geometry. Of course, to really understand algebraic geometry at a deeper level, you will need to acquire some algebraic tools.

If you are a physics student, you most certainly should study abstract algebra, particularly group theory. However, I would adopt an attitude of learning commutative algebra as needed at this point. Depending exactly what level of ability you wish to reach in algebraic geometry will determine how much commutative algebra you need to know. But for a first introduction, focus on the geometric aspects instead of the algebraic aspects, and Griffiths and Harris is a great place to start when this is your goal.

I would highly recommend first covering the book An Invitation to Algebraic Geometry by Smith et al. This requires very little abstract algebra as prerequisite (only understanding of very basics about rings) and gives a very well-written introduction to algebraic geometry. It has a great mix of theory and examples and gives a good idea of what algebraic geometry is and where it goes. As a plus, it has some good exercises asking the reader to work out some examples.