How does the courier work?

I've just gotten into DOTA-2, and never played any similar game before. I realize the courier will bring items out to me, but does every player on my team have their own courier, or is it one per team? If I'm using it to ferry a single item out to me, does that mean it isn't available for other teammates to use? If it is shared, what is the standard courtesy for using it?

Everyone usually shares a single courier that a non-carry player should buy at the beginning and upgrade to flying as soon as possible. You can buy more couriers (as many as you want), but most teams only use one. In general, solo mid gets first rights to courier use, especially in order to get their bottle.

Other than that, it is generally not encouraged for you to bring items to yourself that you could buy at the side or secret shops, as they are close enough you can just walk there from lane. If you do bring something from base, only do so with essential items or "big"(expensive) items. Otherwise wait until you have multiple things to bring out. Ferrying 1 ironwood branch at a time to yourself ties up the courier and will make everyone rage at you.

Furthermore the courier can be used to scout (if flying) but its not recommended since it dies with ~2 hits.

courier can be used to counter neutral spawns too (just like wards). if you get quelling blade/tangos you can eat trees near the enemy jungle pull camp (the camps that you can pull to your lane so creeps attack them) and deploy a courier into that spot.

when the ingametimer reaches 5 min the eaten tree will respawn making the courier hard to find.