Here are four references on the subject (the main ones as far as I know) :
Baker, George A.; Graves-Morris, Peter,
Second edition. Encyclopedia of Mathematics and its Applications, 59. Cambridge University Press, Cambridge, 1996.
Nikishin, E. M.; Sorokin, V. N. Rational approximations and orthogonality. Translations of Mathematical Monographs, 92. American Mathematical Society, Providence, RI, 1991.
Petrushev, P. P.; Popov, V. A.,
Rational approximation of real functions.
Encyclopedia of Mathematics and its Applications, 28. Cambridge University Press, Cambridge, 1987.
Walsh, J. L. Interpolation and approximation by rational functions in the complex domain. Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX American Mathematical Society, Providence, R.I. 1965.
There is also a short chapter on rational interpolation in :
Rivlin, T.J. An introduction to the approximation of functions. Dover Books on Advanced Mathematics. Dover Publications, Inc., New York, 1981.