Drunk is a character condition comparable to tipsy, but much worse. If you continue to drink you will become "totally smashed". You become drunk by drinking multiple alcoholic beverages (available in some Inns, or during special events like Brewfest) in quick succession, without allowing the effects of earlier drinks to wear off.
Your ability to function in World of Warcraft is severely impaired, including random changes of direction while walking/running. Additionally, your typed messages become "drunken"; your "S"'s often become "SH"'s and "...hic!" is added to the end of your lines on occasion (Example: What you shay might come out like thish...hic!).
Getting a character drunk lowers the visual level of an NPC enemy as it appears to the drunken player. The design of this strange effect is speculated to be to trick players into acting more confidently / aggressively, as they might while actually intoxicated.
While the true level of the creature remains the same, it appears lowered to the character who is drunk. The apparent level of a mob goes down by up to 5 levels while "completely smashed", and slowly returns to its actual level while slowly sobering up. The player still receives the same XP for killing mobs as they would while sober. Note that this effect can and will corrupt data collected with addons such as MobInfo2.
It has been theorized that this effect adjusts the resistance level of the creature, making it easier to kill because it resists your spells and blocks/parries/dodges melee attacks as if it was a five levels lower. However, this is very unlikely because of the gameplay advantage it would provide.
Movement is also impaired, as you move sideways quite a lot when moving forwards. Swimming is a bad idea when drunk, as the game makes you swim down when moving forward causing a higher chance to drown, making it more realistic to the real world. Flying is also impaired in the same way as running.
Effects on PvP are unknown.
For supported graphics cards, the more drunk you become the more your screen will blur around the edges until it becomes nearly impossible to see. This can be turned off in the World of Warcraft main menu-->video options by turning off the "full screen glow effect."
In Patch 1.6, Blizzard explicitly added code to stop the "S" to "SH" conversion on certain words that could become profane. Specifically, the word "sit" and its various derivations such as "sits" and "sitting" no longer get translated in this fashion, as long as the profanity filter is turned on.