What is an ideal lattice? Can anyone describe in layman's terms what an ideal lattice is?  I've seen them mentioned in many places, but haven't found a good definition of what exactly they are, nor any good terms to know where to start my search to find more information.
I'm not exactly sure what a non ideal lattice might be either :P
 A: I don't know much about ideal lattices outside the definition, but I can start you off with what a lattice is.  If $n \geq 1$, a lattice in $V = \mathbb{R}^n$ is an additive subgroup of the form $$\Gamma = \mathbb{Z}v_1 + \cdots + \mathbb{Z}v_t$$ where $v_1, ... , v_t$ are linearly independent over $\mathbb{R}$.  If $t = n$, then $\Gamma$ is called a full lattice (many authors mean full lattice when they write lattice).  In other words, a lattice is a free abelian group (that is, an abelian group with a $\mathbb{Z}$-basis), with a basis which is also linearly independent over $\mathbb{R}$.
For example, if $V = \mathbb{R}$, then $\Gamma = \mathbb{Z} + \mathbb{Z}\pi$ is not a lattice, because the $\mathbb{Z}$-basis $1, \pi$ for $\Gamma$ is not linearly independent over $\mathbb{R}$.
But if $V = \mathbb{R}^3$, then $\Gamma = \mathbb{Z}(1,0,0) + \mathbb{Z}(1,0,1)$ is a lattice (of rank two).
Actually, there are several equivalent conditions for an additive subgroup $\Gamma$ of $V$ to be a lattice:
(i) $\Gamma$ is a lattice.
(ii) $\Gamma$ is a discrete subgroup (at every point $x$ in $\Gamma$, you can find an open set in $\mathbb{R}^n$ about $x$ which contains no other point of $\Gamma$)
(iii) Every bounded subset of $\mathbb{R}^n$ intersects $\Gamma$ at most at finitely many points.
You can also talk about quotient groups.  If $\Gamma$ is a lattice in $V = \mathbb{R}^n$, you can talk about the quotient group $V/\Gamma$.  For example, if we take $V = \mathbb{R}$ and the lattice $\Gamma = \mathbb{Z}$, then $V/ \Gamma = \mathbb{R}/\mathbb{Z}$ is really just the circle.  As a topological group, the circle is compact.  More generally, if $\Gamma = \mathbb{Z}v_1 + \cdots + \mathbb{Z}v_t$ for $v_1, ... v_t$ linearly independent over $\mathbb{R}$, you can extend $v_1, ... , v_t$ to a basis $v_{t+1}, ... , v_n$ for $V$.  Then the quotient group $V/\Gamma$ is $$(\mathbb{R}v_1 + \cdots + \mathbb{R}v_n)/(\mathbb{Z}v_1 + \cdots + \mathbb{Z}v_t) \cong \frac{\prod\limits_{i=1}^n \mathbb{R}}{\prod\limits_{i=1}^t \mathbb{Z} \times \prod\limits_{i=t+1}^n \{0\}} $$ $$ \cong \prod\limits_{i=1}^t \mathbb{R}/\mathbb{Z} \times \prod\limits_{i=t+1}^n \mathbb{R} $$ Thus $\Gamma$ is a full lattice if and only if $t = n$, if and only if $V/\Gamma$ is compact.  For detailed proofs, see J.S. Milne's excellent notes on algebraic number theory.
