What does smoothbore mean on these old guns?

Smoothbore means the inside of the barrel has no rifling. The metal tube is simply a straight, polished cylinder from the breech to the muzzle. Most military muskets used this system for centuries. It allowed soldiers to load spherical lead balls quickly in combat, even when the barrel became fouled with black powder residue.

My grandfather left me a hunting shotgun and I was wondering how you can tell what gauge the barrel actually is without any obvious markings?

You measure the internal diameter of the bore at the muzzle using a set of specialist calipers. Antique bore sizes do not always align with modern gauge standards. Gunsmiths historically determined the gauge by the number of solid lead balls of that barrel's diameter it took to make a pound. A twelve bore means twelve round balls to the pound.

Can antique firearms still be fired today?

Many remain capable of firing. The lock springs and steel barrels were built to last. However, old metal develops unseen flaws and black powder generates high pressure. Only a qualified gunsmith can determine if a specific piece is safe to load.

There is a small compartment with a brass lid built into the side of the wooden stock, what on earth was that used for?

That is a patch box. You usually find them on rifles rather than standard military muskets. Shooters kept greased patches of cloth inside. They wrapped these patches around the lead ball before ramming it down the barrel, which helped the ammunition grip the rifling grooves and form a tight seal. Some patch boxes also held spare flints or small tools for field maintenance.