The lattice is really easy to describe. It's a set of points in 8-dimensional space, so each point is specified by 8 numbers. A point is in the lattice if the following conditions hold. (1) Either all its coordinates are integers, or all are half-integers. (2) The sum of the coordinates is an even integer.
(And then the densest packing of spheres in 8-dimensional space puts the centre of a sphere at each point. The radius of the spheres is then 1/sqrt(2).)
The E8 that features in Garrett Lisi's attempted "theory of everything" is a so-called "Lie group". I'll attempt to sketch the relationship between lattices like the E8 lattice, and "Lie algebras", and Lie groups, and particle physics, but note that all of what follows is really handwavy and sometimes just plain wrong. Sometimes on purpose, probably sometimes by mistake. I'm sorry about any mistakes, but the deliberate lies and sketchiness are necessary to keep this shorter than textbook length. I hope it's at least semi-comprehensible.
(Although it's shorter than a textbook, it's also too long for a single HN comment. I'll break it into pieces and make each a reply to its predecessor. HN mods, let me know if this is considered bad form and I'll refrain from doing it in the future.)
OK, here goes.
PHYSICS AND SYMMETRY
OK, let's begin at the physics end. Here are some facts about plain old Newtonian physics. (1) The physical laws make no reference to particular positions in space; equivalently, if we e.g. replace (x,y,z) with (x+a,y+b,z+c) everywhere, for constant a,b,c, then the laws don't change. (2) If you have a system of particles interacting only with one another, the total momentum (sum of mass times velocity) is constant. It turns out that #1 and #2 are sort of equivalent: for a broad class of physical theories, symmetries (like #1, the fact that adding a constant to the coordinates changes nothing) are equivalent to conservation laws (like #2, conservation of momentum). Here's a subtler one: the laws don't pick out preferred times any more than preferred positions, and that fact is equivalent to conservation of energy.
That was all in Newtonian physics, but it works more or less the same in relativity and in quantum mechanics.
There are some much subtler kinds of symmetry present in our physical laws, and they give rise to subtler conservation laws. For instance, you may be familiar with the notion of "phase" in optics; in a classical picture this corresponds to the exact location/timing of the electromagnetic waves that make up light; in quantum mechanics everything is described by a "wavefunction" whose value at every point in the relevant configuration space is a complex number, and the optical phase of light turns out to correspond to the phase of the photons' wavefunctions. Changing all the phases by the same constant amount leaves every possible observation you could make unaltered, which means it's a symmetry of the physical laws; and this symmetry turns out to correspond to conservation of electric charge.
This picture gets more complicated when you throw in the "weak nuclear force", which turns out actually to be kinda the same thing as electricity and magnetism, and hairier still when you throw in the "strong nuclear force". You get symmetries that, e.g., describe exchanging different kinds of particle with one another (kinda), and corresponding conservation laws.
GROUPS AND LIE GROUPS
If you look at a high enough level of abstraction at what different kinds of symmetry have in common, you get the pure-mathematical notion of a group. The idea, roughly, is that if you have some thing (e.g., a block of metal or a physical theory) and some kind of symmetry (e.g., ways you can move the block of metal around and end up with it looking exactly the same as before; ways you can change some things in the laws of physics and end up with equivalent laws), then all the symmetries of the thing together have a particular kind of mathematical structure. This basically comes down to these rather obvious observations: if you have two operations that leave your thing looking the same then doing one and then the other also leaves your thing looking the same; and undoing one such operation also leaves your thing doing the same. And a mathematical structure that supports "doing one thing and then another", and "undoing a thing", is a group.
These symmetries in physics have an important feature: they don't occur in isolation but come in continuously-varying families. E.g., translation-invariance, the (x+a,y+b,z+c) thing, allows any value of a,b,c; you can vary them continuously and get the same results all the time as you do so. This means that the symmetry groups that describe them mathematically have this continuously-varying property, and the mathematical formalization of this gives you the notion of "Lie groups". (Named after a mathematician called Sophus Lie.)
Some Lie groups can be built out of other simpler ones, in something a bit like the way that some numbers can be written as the product of other smaller ones. If you break them down as far as possible, you get to the so-called "simple Lie groups", which are kinda like the prime numbers: every Lie group can be built out of simple Lie groups.
And it turns out that we can figure out what all the possible simple Lie groups are. There are four infinite families of them called A,B,C,D, and a few special ones that don't fit into the families. The most complicated of these special ones is called E8.