An external magnetic field is something that by definition is extraneuous to your object and does not depend on it. This could be a magnet, or the Earth's field itself, to which your sample is exposed to.
It is usually defined by H.
Not sure what exactly your "internal" magnetic field would mean.
The correct terminology to use would be the magnetisation M, of the object, which is the object's own magnetic field (e.g. if it's a magnet), in the absence of any external field.
When you place your magnetic sample in your external field, then the two contributions add up vectorially and you usually use the letter B to describe the outcome
Some objects may have a magnetisation M that emerges from being exposed to an external field H: e.g. a ferromagnetic substance may have all its magnetic domains oriented in random directions therefore adding to 0 net M, but an external field could line them all up in one direction giving a net M $ \neq 0$.
In this case, M and H would have the same direction so B would also point in the same axis. Being just a sum of the two, B could end up being > H (ferromagnets, paramagnets), < H (diamagnets) or $=0$ if M cancels H (superconductors <=> Meissner effect).
H and M are off by a factor of $\mu_0$ wrt to B, because of reasons:
$$ \mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M}).$$
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Usually B is used for magnetic fields everywhere, the distinction betwen B, H and M only really becomes relevant when you have magnetised samples placed in external magnetic fields. The Earth's field is so weak that you usually don't care about it.
