As far as I know, according to google, Eilenberg, Steenrod's book: Foundations of Algebraic Topology was published in 1952, and Spanier's book: Algebraic Topology was published in 1966.
My Questions are the following:
"The first" is either Listing (1847), or Veblen (1922), or Lefschetz (1942), depending on what counts. Here's Dieudonné's book list from his History of Algebraic and Differential Topology:
"It took about 30 years to construct a theory of homology applicable to curvilinear "polyhedra," embodying all the ideas of Poincaré and entirely rigorous... The first treatise on this "classical" algebraic topology was Veblen's Analysis Situs, published in 1922 (but a preliminary version was given as "Colloquium lectures" in 1916); it was followed by the much more complete book Topology by Lefschetz (1930), the very popular Lehrbuch der Topologie of H. Seifert and W. Threllfall (1934), and the book by P. Alexandroff and H. Hopf entitled Topologie I (1935)."
Lefschetz's L'Analysis situs et la géométrie algébrique (1924) can be added to the list. But it should be said that Listing's Vorstudien zur Topologie (1847) already included what came to be a part of algebraic topology (Listing numbers of 2D manifolds). He also pioneered the name "topology" in a letter of 1836.
Betti published his early work on homology in a paper rather than a book, Sopra gli spazi di un numero qualunque di dimensioni (1871), as did Poincare, the official "founder" of the subject, using Leibniz's name Analysis Situs (1892ff). He also coined the term "Betti numbers".
The term "algebraic topology" appears in Steenrod's paper Universal Homology Groups (1936), Lefschetz then used it as the title of his 1942 book, a reworked version of his 1930 Topology, see Earliest Known Uses of Some of the Words of Mathematics. Despite the earlier occurrences, the current spread of "topology" and "algebraic topology" is probably due to his influence, see Markus's biography.
I searched MathSciNet for books with "Algebraic Topology" in the title...
Lefschetz, Solomon Algebraic Topology. American Mathematical Society Colloquium Publications, Vol. 27 American Mathematical Society, New York, 1942.
Eilenberg, Samuel; Steenrod, Norman Foundations of algebraic topology. Princeton University Press, Princeton, N.J., 1952.
Rado, T.; Reichelderfer, P. V. Continuous transformations in analysis. With an introduction to algebraic topology. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXV. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955.
Wallace, Andrew H. An introduction to algebraic topology. Pergamon Press, New York-London, Paris, 1957.
Hilton, P. J.; Wylie, S. Homology theory: An introduction to algebraic topology. Cambridge University Press, New York 1960
Colloquium on Algebraic Topology, August 1–10, 1962. Lectures. Supported by the Scientific Affairs Division of the North Atlantic Treaty Organization Aarhus Universitet, Matematisk Institut, Aarhus 1962
Translations, Ser. 1, Vol. 7: Algebraic topology. American Mathematical Society, Providence, R.I. 1962
Bourgin, D. G. Modern algebraic topology. The Macmillan Company, New York; Collier Macmillan Ltd., London 1963
Narasimhan, M. S.; Ramanan, S.; Sridharan, R.; Varadarajan, K. Algebraic topology. Mathematical Pamphlets, 2 Tata Institute of Fundamental Research, Bombay 1964
Spanier, Edwin H. Algebraic topology. McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966
Hu, Sze-tsen Homology theory: A first course in algebraic topology. Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam 1966
Some other books I feel worth adding are Hilton’s An Introduction to Homotopy Theory and Steenrod’s Topology of Fiber Bundles. Also there are several books on combinatorial topology, like the ones by Alexandroff and Pontryagin, that come after the earlier topology books but still before more “modern” algebraic topology books
