E8 and physics
For well over two decades now String Theory has been the preeminent model for physics beyond the Standard Model. Indeed, String Theory is often touted among physicists as the ultimate Theory of Everything. The basic premise of String Theory is that the most fundamental building blocks of the Universe are not atoms, or even elementary particles like electrons, muons and quarks, but rather exotic string-like objects living in a 26-dimensional space. The great appeal of String Theory to modern physicists is two-fold. First of all, it very deftly circumvents two theoretical obstructions that had long thwarted the unification of Einstein's theory of General Relativity with the quantum field theory of elementary particle physics (viz., the lack of renormalizability and the occurrance of quantization anomalies) . The second is its apparent uniqueness: once one adopts the basic principles of string theory, it can be argued that we live in the universe we live in because it is the only one that is possible.Actually this uniqueness is not quite complete; there are in fact several competing string models. But the dominant model by far is that of heterotic string theory, and it is there that E8 plays an essential role.
Naturally, the most stringent requirement of a viable string theory of the Universe is that eventually the theory has to make contact with the 4-dimensional space-time in which we (at least appear to) live. In heterotic string theory this passage from 26 dimensions to 4 dimensions occurs in two steps. First of all, 16 of the original 26 dimensions must compactify, or curl up on themselves, in a very nice self-consistent way; and then 6 of the remaining 10 dimensions must compactify nicely as well in order to get down to our apparent 4-dimensional observed universe. E8 arises in heterotic string theory because in order for the initial reduction from 26 to 10 dimensions to procede consistently, one needs to endow a 16-dimensional subspace of the orginal 26-dimensional space with an even, unimodular lattice. It turns out that there are exactly two such lattices in 16 dimensions, one of which is the root lattice of E8+E8. See the beautiful exposition by John Baez for more details of this fascinating story.