These recommendation for PDE textbooks are neither meant to be complete or definitive. They are only some books that I have found useful recently while teaching at Harvey Mudd College. Links are to amazon.com which tends to have a great deal more information.

Introductory PDE's for Math Majors

• Applied Partial Differential Equations   by David Logan -- A great introductory textbook available in paperback for about $40.00; while there are more complete texts, they are generally significantly more expensive.
  

• Partial Differential Equations : An Introduction by Walter A. Strauss -- A more complete textbook than Logan, adopted widely for a first course in PDE for math majors. I find it very readable.

Some texts more for the graduate level

• Partial Differential Equations  by Lawrence C. Evans -- A great complete introductory graduate level text
• Introduction to Partial Differential Equations by Gerald B. Folland -- An intermediate graduate level text.
  

On the Applied Side . . .

• Applied Partial Differential Equations by Richard Haberman -- Haberman understands the importance of the applications of PDE without going over to the rather "plug and chug" approach of the engineering texts. A good choice for an introductory course aimed at applied matheticians, physicists, or engineers.

For Engineers only . . .

These two textbooks are aimed at the undergraduate engineer; they tend to cover the whole spectrum of engineering mathematics including an introduction to complex variables, ODEs and some vector calculus.

• Advanced Engineering Mathematics -- by Erwin Kreyszig
• Advanced Engineering Mathematics  -- by Michael Greenberg

Both have been widely adopted.

On the Reference Shelf

• Fourier Analysis by T. W. Körner -- A beautiful, beautiful book with numerous useful examples in Fourier Analysis and PDE, including a number I hadn't seen before (my favorite being the isoperimetric theorem). The writing is engaging and the historic anecdotes are priceless.
• Fourier Analysis and Its Applications by Gerald B. Folland -- An advanced undergraduate/introductory graduate text on Fourier analysis and distributions with applications to functional analysis and PDE.
• Introduction to Fourier Analysis  by M. J. Lighthill -- A marvelous book --while written in 1958, it has one of the most coherent introduction to the theory of distributions from the applied viewpoint I have seen.
• Mathematical Physics by Eugene Butkov -- A great source for problems germane to the physicist. While I tried this as a textbook and found it too terse for undergraduate, it still has a place on my bookshelf as a reference in the field.