Stanley [4] stated that "the basic problem of enumerative combinatorics is that of counting the number of elements of a finite set."

More later - a brief history and general overview.

Permutations, words, matrices
Factorials, binomial coefficients, combinatorial functions
Exact enumeration problems, generating functions
Asymptotic enumeration
Partitions of integers
Partitions of sets
Combinatorial identities, bijective combinatorics
Combinatorial inequalities
q-calculus and related topics
Umbral calculus
Other Areas of Enumerative Combinatorics

Books:

[1] Bóna, Miklós Introduction to enumerative combinatorics. With a foreword by Richard Stanley. Walter Rudin Student Series in Advanced Mathematics. McGraw Hill Higher Education, Boston, MA, 2007. xvi+526 pp. ISBN: 978-0-07-312561-9; 0-07-312561-X

[2] Charalambides, Charalambos A. Enumerative combinatorics. CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002. xvi+609 pp. ISBN: 1-58488-290-5

[3] Martin, George E. Counting: the art of enumerative combinatorics. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2001. xii+250 pp. ISBN: 0-387-95225-X

[4] Stanley, Richard P. Enumerative combinatorics. Volume 1. Second edition. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 2012. xiv+626 pp. ISBN: 978-1-107-60262-5

[5] Stanley, Richard P. Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. xii+581 pp. ISBN: 0-521-56069-1; 0-521-78987-7

Online Lecture Notes:

[1] Enumerative Combinatorics Through Guided Discovery (Kenneth P. Bogart, Dartmouth)

[2] Notes on Counting: An Introduction to Enumerative Combinatorics (Peter J. Cameron, University of London)

[3] Enumerative Combinatorics The LTCC lectures (Peter J. Cameron, University of London)

[4] Lecture notes for “Enumerative Combinatorics” (Anna de Mier, University of Oxford)

Online Video Lectures:

[1] Lecture 1 . Enumerative Combinatorics (Federico Ardila)

[2] Lecture 2 . Enumerative Combinatorics (Federico Ardila)

[3] Lecture 3 . Enumerative Combinatorics (Federico Ardila)

[4] Lecture 4 . Enumerative Combinatorics (Federico Ardila)