Geodesic incompleteness occurs when a space has "holes" in it, or if it is possible to "fall off the edge" of the space. Here are some examples:
The punctured plane $\mathbb{R}^2 - \{(0,0)\}$ is geodesically incomplete in the standard metric. For example, a horizontal geodesic starting at $(-1,0)$ and traveling in the positive $x$-direction can only be extended for one unit.
Note that this space is homeomorphic to an infinite cylinder $S^1\times\mathbb{R}$, which is complete under the standard metric. The difference is that it isn't possible to get to infinity in a finite distance on the cylinder, while it is on the punctured plane.
Covers of a geodesically incomplete space are also geodesically incomplete. Thus all of the covers of the punctured plane are geodesically incomplete, including the universal cover.
The open unit disc $\{(x,y)\in\mathbb{R}^2 \mid x^2+y^2 < 1\}$ is geodesically incomplete in the standard metric. No geodesic can be extended for more than $\pi$ units.
The open upper half plane $\{(x,y)\in\mathbb{R}^2\mid y >0\}$ is geodesically incomplete in the standard metric. As with the disc, the idea is that geodesics can fall off the edge of the space.
Note that the upper half plane is complete under the hyperbolic metric. The difference is that it takes infinite time to reach the $x$-axis under the hyperbolic metric, while it can be reached in finite time under the Euclidean metric.
Most of the examples so far are proper open subsets of complete Riemannian manifolds, but not every geodesically incomplete space has this form. For example, consider the infinite cone $C=\{(x,y,z)\in\mathbb{R}^3 \mid x^2+y^2=z^2\text{ and }z>0\}$ under the standard metric, and note that the vertex $(0,0,0)$ does not lie in $C$. This space is incomplete because geodesics can approach the point $(0,0,0)$, but it doesn't work to add this point because the closed cone $C\cup\{(0,0,0)\}$ isn't a Riemannian manifold.
The covers of the punctured plane mentioned in example #1 also aren't open subsets of complete Reimannian manifolds.
The most basic way to prove that a Riemannian manifold $M$ is incomplete is to identify a geodesic $\gamma\colon[0,a)\to M$ that cannot be extended any further. The way to prove that $\gamma$ cannot be extended is to show that $\lim_{t\to a} \gamma(t)$ does not exist in $M$.
By the Hopf-Rinow theorem, you can also prove that $M$ is geodesically incomplete by proving that $M$ is incomplete as a metric space. Thus, you can also show that $M$ is geodesically incomplete by demonstrating a Cauchy sequence that doesn't converge, or by demonstrating that some closed and bounded subset of $M$ in not compact.