The Emperor's body is physically about as dead as you can get, but he is such a powerful Psyker (read: magic user) that they keep his body alive on life support in order to harness the powers of his brain. The life support system (Called the Golden Throne) is so demanding that they literally need to provide him ~1000 human sacrifices per day, specifically people with psionic/Psyker capabilities.

However, a huge amount of the Empire's logistical infrastructure depends on him, since his mind provides a beacon (called the astronomicon) that allows ships to navigate the Warp (another dimension, full of Eldritch horrors, useful for FTL travel, similar to D&D's astral plane).

Most of the empire has become convinced that the Emperor himself is a living God, that he is omniscient and can enact his will around the galaxy. They are somewhat justified in this as well, since praising the Emperor is an effective way of repelling demons (IIRC this is just because believing in anything with enough zeal will repel demons).

Also, as soon as someone begins to doubt the divinity of the emperor, the chaos gods are able to start influencing/corrupting them, so it's generally considered a dick move not to praise the emperor. Agnosticism, atheism, and even just having your own interpretation about the imperial cult literally represent a threat in the same way a terrorist might.