The Sylow theorems are finite group analogues of a bunch of results about "maximal unipotent subgroups" in algebraic groups. Basically, the Sylow subgroups play a role analogous to the role played by the maximal unipotent subgroups. 

In the case where the group is the general linear group, the maximal unipotent subgroup can be taken as the group of upper triangular matrices with 1s on the diagonal, for instance. There are existence, conjugacy, and domination results for these analogous to the existence, conjugacy, and domination part of Sylow's theorems: maximal unipotents exist, every unipotent is contained in a maximal unipotent, all maximal unipotents are conjugate. The role analogous to "order" is now played by "dimension".

The normalizer of the Sylow subgroup plays the role of the maximal connected solvable subgroup, also called the Borel subgroup (see <a href="http://en.wikipedia.org/wiki/Borel_fixed-point_theorem">Borel fixed-point theorem</a> and <a href="http://en.wikipedia.org/wiki/Lie-Kolchin_theorem">Lie-Kolchin theorem</a>). In the case of the general linear group, this is the group of upper triangular invertible matrices.

There are similar results for Lie algebras too, basically arising from <a href="http://en.wikipedia.org/wiki/Engel_theorem">Engel's theorem</a> and <a href="http://en.wikipedia.org/wiki/Lie%27s_theorem">Lie's theorem</a>.

In fact, much of the study of simple groups and their geometry relies on this geometric interpretation of Sylow subgroups, <em>p</em>-subgroups, and their normalizers. This deeper study of the geometry/combinatorics of simple groups is called local analysis in group theory and is closely related to the recently popular topic of "fusion systems" which are essentially studying the conjugation action of a group on subgroups of a particular Sylow subgroup.