Revival can happen for both you and your minion. Upon complete death, reincarnation is always an option for Red. The reasons behind this are in the plot, and will not be mentioned here for the sake of spoilers, although they may be found in other pages.
Reincarnation allows you to keep your lore knowledge, as well as any Overworld and Underworld Portals on the same floors they were previously, but you lose all your levels and gear (although gear dropped in the overworld will remain). Basically, knowledge is stored within you, but the quickenings and equipment were of the Dungeon of Chaos, which runs on its own time. The Overworld is "outside of the chaotic time" and the Underworld is "at the end of time," so both remain even when you reincarnate.
The chest on the Overworld will also stay the same: opened if you already opened it, and closed if you haven't yet. This is why it is recommend only opening it when you're ready to make a serious attempt at beating the game.
Reviving pre-reincarnation for the rogue is done by eating a Rune of Undead while you are a non-Undead race, like Rogue or Troll. If you eat it, you will survive death by coming back as a Skeleton. Since the Overworld is outside of time, returning to the Overworld grants you your Rogue status again, which has higher stats. This is very useful and can be used in multiple ways.
Eating a Rune of Undead while being Undead just turns it into a healing item.
Reviving the Minion
The Minion is normally Undead, but with Faces or Polymorphs can become a living race, in which case you can use the Rune of Undead in the same way to grant it an extra life.
As an Undead, when the Minion dies, it immediately is sent to the Underworld. If you are near an Underworld Portal, it will come out and rejoin your party. It is best to go back to the nearest Underworld Portal as soon as possible to get the Minion again, because he is incredibly helpful.
If you aren't near an Underworld Portal, you can throw a Rune of Undead at a non-Undead enemy and kill it. That sends their soul to the Underworld via Underworld Portal, and causes an Underworld Portal to open. The Rune of Portal should also be able to generate an Underworld Portal, but the portal is not always going to be an underworld portal.
You can't revive the minion clones given by RNG, unless you get more from him, which technically isn't revival.
By entering the Konami code, you can gain extra lives where you don't lose your levels and gear. This is a form of cheating, and may be intended as a joke, easter egg, or debugging tool not meant for practical use.
At any rate and in order, press the following in order to gain a life.
Up Up Down Down Left Right Left Right B A
Per game, this can only be used three times to gain three lives before the game calls you out for cheating.