Friendly Introductions:

Hatcher, Allen Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544

By far the most popular introduction. Friendly casual style. Many good problems. Cons: Would be better to leave out $\Delta$ complexes.

Matveev, Sergey V. Lectures on algebraic topology. Translated from the 2003 Russian original by Ekaterina Pervova and revised by the author. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), ZŸrich, 2006. viii+99

Vick, James W. Homology theory. An introduction to algebraic topology. Second edition. Graduate Texts in Mathematics, 145. Springer-Verlag, New York, 1994. xiv+242

Short concise introduction

Greenberg, Marvin J.; Harper, John R. Algebraic topology. A first course. Mathematics Lecture Note Series, 58. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. xi+311

Massey, William S. A basic course in algebraic topology. Graduate Texts in Mathematics, 127. Springer-Verlag, New York, 1991. xvi+428

Best place to be introduced to covering spaces and fundamental groups. Many nice applications.

Bredon, Glen E. Topology and geometry. Corrected third printing of the 1993 original. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1997. xiv+557

Includes manifold theory.

Munkres, James R. Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984. ix+454

Homology of simplicial complexes.

Maunder, C. R. F. Algebraic topology. Reprint of the 1980 edition. Dover Publications, Inc., Mineola, NY, 1996. viii+375

Dover books are very cheap.

Kosniowski, Czes A first course in algebraic topology. Cambridge University Press, Cambridge-New York, 1980. viii+269

Friendly Introductions via Differential Forms:

Fulton, William Algebraic topology. A first course. Graduate Texts in Mathematics, 153. Springer-Verlag, New York, 1995. xviii+430

Great at explaining connections to other area of mathematics. (Fulton is a very distinguished algebraic geometer.) However it does not cover many of the standard topics.

Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982. xiv+331

Bott has proved some of the deepest most important theorems in topology.

Madsen, Ib; Tornehave, J¿rgen From calculus to cohomology. de Rham cohomology and characteristic classes. Cambridge University Press, Cambridge, 1997. viii+286

Friendly Introduction via Stratified Spaces:

Kreck, Matthias Differential algebraic topology. From stratifolds to exotic spheres. Graduate Studies in Mathematics, 110. American Mathematical Society, Providence, RI, 2010. xii+218 pp

Harder Introductions:

Spanier, Edwin H. Algebraic topology. McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966 xiv+528

May, J. P. A concise course in algebraic topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1999. x+243

Modern advanced viewpoint. May and Ponto will soon be publishing Part II which contains many important advanced topics

tom Dieck, Tammo Algebraic topology. EMS Textbooks in Mathematics. European Mathematical Society (EMS), ZŸrich, 2008. xii+567

Dold, Albrecht Lectures on algebraic topology. Reprint of the 1972 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. xii+377

Very strong on fixed point theory.

Gray, Brayton Homotopy theory. An introduction to algebraic topology. Pure and Applied Mathematics, Vol. 64. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. xiii+368

This book first gives spectral definition of homology and cohomology. Chain complexes are not introduced until middle of the book.

More Advanced:

Studies in modern topology. P. J. Hilton, editor. Studies in Mathematics, Vol. 5 Published by the Mathematical Association of America; distributed by Prentice-Hall, Inc., englewood Cliffs, N.J. 1968 vii+212 pp

This is a very nice nice collection of survey articles

Davis, James F.; Kirk, Paul Lecture notes in algebraic topology. Graduate Studies in Mathematics, 35. American Mathematical Society, Providence, RI, 2001. xvi+367

Includes homology and cohomology with twisted coeffients and spectral sequences.

Aguilar, Marcelo; Gitler, Samuel; Prieto, Carlos Algebraic topology from a homotopical viewpoint. Translated from the Spanish by Stephen Bruce Sontz. Universitext. Springer-Verlag, New York, 2002. xxx+478

Mosher, Robert E.; Tangora, Martin C. Cohomology operations and applications in homotopy theory. Harper & Row, Publishers, New York-London 1968 x+214

Switzer, Robert M. Algebraic topologyÑhomotopy and homology. Reprint of the 1975 original [Springer, New York; MR0385836 (52 #6695)]. Classics in Mathematics. Springer-Verlag, Berlin, 2002. xiv+526

Rudyak, Yuli B.(D-HDBG) On Thom spectra, orientability, and cobordism. With a foreword by Haynes Miller. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xii+587

Frank Adams on Advanced Topics : (Highly Recommended)

Stable homotopy and generalised homology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, Ill.-London, 1974. x+373

Localisation and completion. With an addendum on the use of Brown-Peterson homology in stable homotopy. Lecture notes by Z. Fiedorowicz on a course given at the University of Chicago in Spring, 1973. Department of Mathematics, University of Chicago, Chicago, Ill., 1975. ii+141

Prerequisites (on equivariant stable homotopy) for Carlsson's lecture. Algebraic topology, Aarhus 1982 (Aarhus, 1982), 483Ð532, Lecture Notes in Math., 1051, Springer, Berlin, 1984

Lectures on generalised cohomology. 1969 Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three) pp. 1Ð138

Adams, John Frank Infinite loop spaces. Annals of Mathematics Studies, 90. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. x+214 pp

Adams, John Frank Algebraic topologyÑa student's guide. London Mathematical Society Lecture Note Series, No. 4. Cambridge University Press, London-New York, 1972. vi+300

John Milnor on Geometric Topics: (Highly Recommended)

Milnor, John W.; Stasheff, James D. Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. vii+331 pp

Milnor, John Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow Princeton University Press, Princeton, N.J. 1965 v+116 pp

Milnor, J. Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963