What sort of weapon fires this small explosive round?

From this CNN report.

This video is footage of an AK-47 shooting HE ammo. That would be the most likely suspect if the round came from a non-US gun.

I am NOT a gun expert (IANAGE)

This article says it was a 14.5mm round. Kinda close to a .50 BMG round. Something for a machine gun.

That looks like a pretty big bullet – definitely bigger than the 7.62 mm AK-47 round. It’s probably a bigger round, 12.7 mm or larger round of Russian vintage. That sort of round is used in a heavy machine gun or small cannon, which commonly use explosive bullets.

ETA: Beaten by Hermitian!

Damn, that’s bigger than I thought it was. Definitely from a machine gun or something similar.

The 14.5 was a standard USSR light anti-aircraft cannon. See http://fas.org/man/dod-101/sys/land/row/zpu-1.htm . The same basic breech / barrel asembly has been repackaged into lots of different systems since the original ZPU-1. Including mounting it on light armored vehicles intended to engage other thin-skinned ground targets.
FYI, by definition, a “machine gun” shoots non-explosive projectiles, while a “cannon” shoots explosive projectiles. At least this is the modern US DOD definition of the terms.