Specific examples of Eilenberg-Maclane spaces?

For $K(G,1)$ spaces there are some geometric methods. The key thing about the $K(G,1)$ property is that a connected CW complex possesses that property if and only if the universal covering space of that complex is contractible. And there are several examples of theorems of the form "such-and-such hypotheses imply that the universal covering space is contractible".

Here's one such class of examples, coming from Riemannian geometry. Let $M$ be a smooth connected $m$-manifold, and let $g$ be a complete Riemannian metric on $M$ (for what it's worth, all smooth compact manifolds have finite CW-complex structures). If all sectional curvatures of $g$ are non-positive, then $M$ is a $K(G,1)$ space. The reason this is true is because the universal cover $\widetilde M$ is simply connected $m$-manifold, the lift $\tilde g$ is a complete Riemannian metric with nonpositive sectional curvatures, and now one applies the Cartan-Hadamard theorem to conclude that $\widetilde M$ is diffeomorphic to $\mathbb R^m$ and is therefore contractible, and so $M$ is a $K(G,1)$ space.

For some very specific examples, if $m=2$ and if $M$ is any connected 2-manifold that is not homeomorphic to $\mathbb S^2$ or $\mathbb R P^2$ then $M$ is a $K(G,1)$ space (because all such surfaces have a complete Riemannian metric of constant curvature $0$ or $-1$, by an application of the Riemann mapping theorem). So, for example, for each $g \ge 2$ the closed, oriented surface of genus $g$ is a $K(G,1)$ space.