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Roughly speaking, a finite set together with a generalization of a concept from linear algebra that satisfies a natural set of
properties for that concept. For example, the finite set could be the rows of a Matrix, and the generalizing concept
could be linear dependence and independence of any subset of rows of the Matrix. The number of matroids with 
points
are 1, 1, 2, 4, 9, 26, 101, 950, ... (Sloane's A002773).
References
Sloane, N. J. A. Sequence
A002773/M1197
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and extended entry in
Sloane, N. J. A. and Plouffe, S.
The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
Whitely, W. ``Matroids and Rigid Structures.'' In Matroid Applications, Encyclopedia of Mathematics and Its Applications
(Ed. N. White), Vol. 40. New York: Cambridge University Press, pp. 1-53, 1992.