I saw the definition of a geodesic sphere, and I think I'm not able to "see" how do they look like. For example, it's obvious that in $\mathbb R^n$ geodesic spheres are simply normal spheres, and that in $\mathbb S^n$ they are intersection of $\mathbb S^n$ with a little sphere $S$ centered at $p$ giving $\mathbb S^{n-1}$ by a symmetry arguments.
What about in other Riemannian manifolds, for example $\mathbb H^2$ ?
Geodesic spheres in hyperbolic space are isometric to Euclidean spheres. The catch is, if the ambient curvature is $-1$ and the geodesic sphere $S$ has hyperbolic radius $r$, then $S$ is isometric to a Euclidean sphere of radius $\sinh r$. (Analogously, if $0 < r < \pi$, then a geodesic sphere in the round unit sphere $S^{n}$ is intrinsically isometric to a Euclidean sphere of radius $\sin r$.)
If you look at geodesic spheres in a conformal model of the hyperbolic plane (the upper half-plane or Poincaré disk), a geodesic sphere of radius $r$ is a Euclidean circle (of center-dependent radius smaller than $r$). The hyperbolic center is not generally the Euclidean center. (The exception is circles centered at the origin in the disk model.)
Incidentally, a hyperbolic circle of hyperbolic radius $r$ centered at the origin in the disk model has "extrinsic" Euclidean radius $\tanh r$. Note carefully that this means a single sphere has three radii: Its intrinsic hyperbolic radius $r$, its intrinsic Euclidean radius $\sinh r$, and the actual (extrinsic) radius, $\tanh r$, that you draw in the disk model if the center is the origin.