⋅ is the infamous "multiplying dot" of mathematical typography. Don't confuse it ("⋅") with the lesser central dot ¸ that is "¸"; this is the real thing. LaTeX calls it "\cdot", which is confusing when compared to the HTML entity names. Maybe the other one could be used for the raised decimal point sometimes used instead of the period?
Multiplication is an odd beast, starting roughly from seventh grade. Writing out "a×b" is declassé and Not Done. Usually, the entire operator is dropped, and we have "ab". But sometimes "a⋅b" is used instead. Since omitting it is common, leaving it in emphasises it. For instance, one might write
(a+b)&sdot(a-b) = a2-b2.
When dealing with non-scalar elements, typically vectors but also matrices and the like, the "dot product" is commonly used for the inner product, aka scalar product or Hermitian product of a vector space. Contrast with the "cross product" that is ×, or the weird and wonderful "tensor product" that is ⊗. The various forms of matrix multiplication are also dotted (or left out entirely); the "rule" seems to be that the dot lowers the dimension of the result compared to the operands.
Yet other places where you sometimes might see a "⋅" hiding involve groups, usually non-abelian ones.
The multiplying dot also has its problems, though. Since a central dot can be used for a decimal point, it is never used when both operands are numeric. You'll never see "7⋅9" for "63", as it could (almost) also be what some prefer to call "7.9".