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.
This post imported from StackExchange MathOverflow at 2017-05-19 14:21 (UTC), posted by SE-user Alexandre Eremenko