Our goal here is to do geometry. If we only cared about the topology, given two points on a manifold, I can stretch the two points out to appear seemingly far apart, or I could squeeze the space between them, making them appear close togeether. There is no well defined notion of distance between points in a manifold, so we need something else. When we first learn geometry, we computed the distance between two points in $\mathbb{R}^3$ by looking at the length of a straight line connecting them. So in order to make sense of distance, we need to make sense of lengths of curves. Let's think back to Calc 3, how did we compute the length of a curve in $\mathbb{R}^3$? We used the fact that if you integrate an infitesimal velocity, you obtain an infitesimal measure of length of a curve, thus what you need to do in order to measure length is to integrate the velocities along the curve. Namely for a curve $\gamma: [a, b] \rightarrow \mathbb{R}^3$, you get $$ L(\gamma) = \int_a^b \sqrt{\left< \gamma'(t), \gamma'(t)\right>}dt $$ Of course, the reason why we can do this is because in the usual $\mathbb{R}^3$ from calculus, we have a natural way to measure the size of a vector, namely the square root of the usual inner product. So in order to compute the sizes of velocity vectors of curves on manifolds we need two ingredients; velocity vectors of curves and an inner product.
To address the first point, the collection of velocities of curves passing through a given point $p \in M$ is simply the tangent space denoted $T_pM$. The origin of this name can be seen by thinking back to computing the velocities of curves on a sphere in $\mathbb{R}^3$. Such vectors are tangent to the surface, and form a plane which meets the sphere tangentially. Now in order to clarify the second point, we simply need to smoothly assign an inner product to each tangent space of the manifold $M$. We call a smooth assignment of inner products to the tangent spaces of a manifold a Riemannian metric.
A manifold $M$ equipped with a Riemannian metric is called a Riemannian manifold. One can show (in a usual differential geometry course) that any smooth manifold admits a Riemannian metric, and additionally that there are infinite dimensional affine space of choices for metrics on a given manifold, thus it may not be obvious which metric to use. Fortunately, Lie groups have natural choices of metrics. Recall that a group acting on itself produces a family of bijections. $ h \mapsto gh$ is a bijection for any $g \in G$. Recall that the group operation is smooth, thus the group operation produces a diffeomorphism. Now since every element has an inverse, we can then take any point, and find a natural diffeomorphism which sends it to the identity element. What we then do is assign an inner product to $T_eG$ and then for vectors $v, w \in T_p G$, we can find the diffeomorphism $\Phi: G \rightarrow G$ such that $\Phi(p) = e$ and then $d\Phi : T_p G \rightarrow T_e G$ will map $v$ and $w$ to vectors in $T_eG$, then $$ \left< v, w \right>_p = \left< d\Phi_p(v), d\Phi_p(w)\right> $$
Now let's return to $\mathcal{H}$ and assign the usual inner product to $T_e\mathcal{H}$. Given a point in $\mathcal{H}$ we need to understand what the inner product will be.
$$ ds^2 = dx^2 + dy^2 + (dz - x dy)^2 $$ thus given $v, w \in T_p \mathcal{H}$, $$ g(v, w) = v^T \begin{pmatrix} 1 & 0 & 0 \\ 0 & (1+x^2) & -x \\ 0 & - x & 1 \end{pmatrix} w $$ Now since we've defined this metric to be invariaant under the left action by elements in $\mathcal{H}$, it follows that all of the diffeomorphisms (of left action) are upgraded to isometries! This gives the group structure of $\mathcal{H}$ (and indeed any Lie group assigned a left invariant metric) geometric weight.Why is this geometry called Nil? The reason is because the model space, namely the Heisenberg group, is a nilpotent group. A group $G$ is said to be nilpotent if the following sequence $[G,G], [G, [G,G]], [G,[G,[G,G]]], \dots$ terminates with $\left< e \right>$. An equivalent formulation of nilpotency is that the following ascending central series $C_0(G) = e$, $C_1(G) = Z(G)$, $C_2(G) = p^{-1}(Z(G/Z(G)))$, $\dots$, terminates at $C_n(G) = G$. We can see that the center of the Heisenberg group consists of elements: $$ Z(\mathcal{H}) = \left\{ \begin{pmatrix} 1 & 0 & z \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} | z \in \mathbb{R} \right\} $$ Clearly $\mathcal{H}/ Z(\mathcal{H}) = \mathbb{R}^2$ which is abelian, and thus $Z(\mathcal{H}/Z(\mathcal{H})) = \mathcal{H}/Z(\mathcal{H})$, the preimage of which is the entire Heisenberg group.
The next question is how might Nil look like to someone living inside? In order to find out we need to understand some properties of light. Light will always travel along straight paths, or paths which locally minimize curvature and arclength. Recall that we had a notion in calculus of how to compute the curvature of a curve, namely the acceleration. We took the derivative of the tangent vector of a curve in the direction the curve was heading (with respect to arclength). In a manifold equipped with a metric we can do something similar, only now we have to keep track of the new metric structure. The equation then for the curving of a curve is the following: $$ K = \nabla_{\gamma'(t)} \gamma'(t) $$ and since we are interested in straight paths, we want the following equation: $$ 0 = \nabla_{\gamma'(t)} \gamma'(t) $$ which is called the Geodesic equation since curves which take the straightest path are called geodesics. In coordinates this expression takes the form $$ \frac{d^2x^k}{dt^2} + \sum_{i,j} \Gamma^k_{i,j} \frac{d x^i}{dt} \frac{dx^j}{dt} $$ where $\Gamma^k_{i,j}$ are coefficients derived from the metric commonly called Christoffel symbols. The formula to compute them is a little unweildy: $$ \Gamma^k_{ij} = \frac{1}{2} g^{k m} ( g_{im,j} + g_{jm, i} - g_{ij,m}) $$ where $g_{ij, k}$ denotes the derivative of the $ij$th entry of the metric with respect to the $k$th coordinate.
Now that we've extensively talked about the Heisenberg group and its geometry, we can talk about spaces which are modeled on Nil geometry. A model geometry is a manifold $M$ together with a Lie group $G$ which acts transitively on $X$, and such that the stabilizers are compact. We think of the Lie group acting as the group of isometries on $M$. Let $\Gamma$ be a discrete subgroup of $G$, then $N = M / \Gamma$ which is the spcae obtained by quotienting via the action of $\Gamma$ is a space modelled on $(M, G)$. And it possesses a metric inheritted from $G$ which retains much of its features. a Nil manifold is a manifold modelled on $\mathcal{H}$.