If my enchanted creature dies, and I then bring the creature back to the battlefield does said creature come back with the same enchantment on it?
ConMan's answer is very good, but it goes explaining auras, their effects and characteristics but I think this rule is more important to explain the answer to your question.
217.1c An object that moves from one zone to another is treated as a new object. Effects connected with its previous location will no longer affect it.
Answer to your question: No, it will not have any auras or equipment attached to it.
This is because every time an object moves from one zone to another, this is a new object. Whether this new zone is graveyard, battlefield, exile, library or hand.
A good and relevant example of this is a card by the name of Animate Dead with the following text:
Enchant creature card in a graveyard When Animate Dead enters the battlefield, if it's on the battlefield, it loses "enchant creature card in a graveyard" and gains "enchant creature put onto the battlefield with Animate Dead." Return enchanted creature card to the battlefield under your control and attach Animate Dead to it. When Animate Dead leaves the battlefield, that creature's controller sacrifices it.
Very convoluted text for a simple, return creature to the battlefield isn't it? The gatherer explains why:
Animate Dead is an Aura, albeit with an unusual enchant ability. You target a creature card in a graveyard when you cast it. It enters the battlefield attached to that card. Then it returns that card to the battlefield, and attaches itself to the card again (since the card is a new object on the battlefield). Animate Dead itself never moves into a graveyard during this process.
As Animate Dead attaches to the creature and enters the battlefield, the creature is still theoretically in the graveyard, the effect from Animate Dead then brings the creature to the battlefield which means that as it is now a new object, Animate Dead has to attach itself to it again.
From the Comprehensive Rules (note that an Aura is an enchantment that attaches to another permanent):
303.4c If an Aura is enchanting an illegal object or player as defined by its enchant ability and other applicable effects, the object it was attached to no longer exists, or the player it was attached to has left the game, the Aura is put into its owner’s graveyard. (This is a state-based action. See rule 704.)
So when the creature leaves play and is no longer a valid target for the Aura, the Aura is put into its owners graveyard. If the creature is later returned to the battlefield, the Aura has already forgotten that it was attached to the creature and does not come back.
The one exception to this would be if the creature regenerates:
701.14a If the effect of a resolving spell or ability regenerates a permanent, it creates a replacement effect that protects the permanent the next time it would be destroyed this turn. In this case, “Regenerate [permanent]” means “The next time [permanent] would be destroyed this turn, instead remove all damage marked on it and tap it. If it’s an attacking or blocking creature, remove it from combat.”
Note that regeneration causes an effect that replaces the destruction of the creature, so it never actually goes to the graveyard, and hence the Aura stays attached to it.
EDIT: As pointed out, Totem Armor is another related situation:
702.88a Totem armor is a static ability that appears on some Auras. “Totem armor” means “If enchanted permanent would be destroyed, instead remove all damage marked on it and destroy this Aura.”
In this case, if the creature is enchanted with an Aura that has this ability, then the Aura gets destroyed instead of the creature. So here the creature does not get destroyed, but the Aura does.
No, the creature will not be enchanted.
There are a variety of reasons for this, but the most basic reason is that the a resurrected card doesn't know it was ever previously enchanted. And cards only do as much as they say, so the resurrection spell won't enchant the creature unless it says otherwise.
According to the rules (my notes)
400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence. There are seven exceptions to this rule (none of the exceptions apply)