Matroids
A matroid is a combinatorial structure that generalizes the properties of dependence and independence exhibited by sets of vectors in vector spaces. Vector spaces are exceedingly useful, partly because properties such as every basis having the same number of elements and any independent set being extendable to a basis make them very easy to manipulate. However vector spaces rely absolutely on the existence of a field which is an algebraic structure, and these useful properties follow directly from the properties of fields.
Matroids are the structures that arise when we require only that these useful properties hold, regardless of whether the structure actually is a vector space or not. What is genuinely surprising is that almost no matter how we express these properties, the exact same class of structures results!