The efficacy of armor against arrows varies substantially with the circumstances involved. Type of armor, type of arrow point, range, bow poundage, etc. That said, typically plate armors like lorica segmentata (the banded one that is usually depicted in pop culture) would fare better than other things but they are in no way proof against all projectiles. Maille/lorica hamata is often held to be susceptible to arrows yet depending on the subarmalis and where it hits, certain arrows can have a hard time penetrating. So to sum up, it really depends.

As for taking a single arrow and immediately dying, that's really more of a Hollywoodism. Pretty much the only way a person is going to immediately keel over from an arrow is if it's right in the eyes/brain or directly into the heart. Heck even someone that gets tagged in the jugular or femoral is going to have at least a few seconds of complaining before going down. Elsewhere and the guy is probably going to keep chugging along because lying down isn't really an option in a fight. There are stories from the Crusades that tell of men pierced by several arrows yet continuing to fight and win. The stories the men told and the way they fought while wounded in Kingdom of Heaven was inspired by such stories. In the modern age you can look at how wounded Soldiers get, yet they can and do fight on if the situation requires. Lone Survivor is Hollywooded too some but is still a fairly good depiction of how much damage a body can sustain but keep on fighting. Bowden's Black Hawk Down (the book, the movie glossed over a lot of details) details how torn up the Soldiers got in the battle of Mogadishu, to the point where almost everyone in the fight was wounded in some way, especially in the lost convoy. Adrenaline is awesome.