It's not necessarily the path of shortest distance. For example, if you have two points on a sphere, there are two (non-self-intersecting) geodesics joining them, one of which goes 'the short way round' the sphere and the other of which goes the long way round.
You're asking quite a big question though. If you're just after definitions then why not look them up?
I'm trying to understand different types of geodesics. I understand that a geodesic is the shortest distance between two points on a manifold, but what's are these types:
metric - I assume this is just the distance as calculated from the metric tensor
Can someone explain these?
The wikipedia isn't too bad: let me summarize briefly what I learned from 5 minutes reading having previously known dirt all about (Psuedo)-Riemannian geometry:
A (metric) geodesic on a Riemannian manifold is loosely a curve that is locally the shortest distance between two points.
If you have a smooth manifold with an affine connection then one can define a geodesic of a curve in terms of parallel transport. On a Riemannian manifold, if you take the Levi-Civita connection and solve the geodesic equations then you can show that a geodesic in this sense is the same as a geodesic in the sense of locally minimising the metric.
The last three adjectives, as one would expect, refer to geodesics on a Lorentzian manifold. One may also equip a Lorentzian manifold with the Levi-Civita connection which is the unique connection satisfying the same properties as in the Riemannian case. Thus again there is no distinction between metric geodesics and affine geodesics w.r.t Levi-Civita. Then (assuming the positive part of the metric is space and the negative part is time) a geodesic is space like if the norm of its tangent vector is positive, timelike if the norm of its tangent vector is negative and null if it is zero.