There is a big differences between dungeons and raids.
Dungeons (whether they be normal or heroic) are designed for 5 people to complete. Raids are designed for 10 or 25 people to beat (there are also old 20/40 man raids as well, and in the future the hardest raid difficulty will be 20 people only). Dungeons are supposed to be a stepping stone to get into raids. They drop lower ilvl loot, and are mechanically easier on the player.
For example, in this current MoP expansion, Heroic Dungeons drop ilvl 463 loot. In comparison, the first raiding tier (in this case it is tier 14) drops ilvl 489/496 in normal modes, and 502/509 in heroic modes. The second raiding tier (t15) drops 522 in normal and 535 in heroic, and the third raiding tier (t16, the current raiding tier) drops 553 in normal and 566 in heroic.
Ilvl stands for Item Level. It is what determines how many stat points the item can have on it (in terms of how much of a particular stat it gives, like how much agility or how much crit is on it). There is a large difference in even a few ilvls, so that if you had a raid of people in full 580+ gear, you would be able to waltz thru t14 with your eyes closed, significantly fewer healers, and kill bosses much faster with much higher dps.
And that's all in the same expansion, where you're the same level.
Consider previous expansions, where you gain not just the ilvl boost, but also simply being higher level. For example, The Lich King at the end of Icecrown Citadel drops gear that ranges from 258 on 10man normal (back then 10man dropped different, lower ilvl loot but was also on a different lockout from 25man) and 284 on heroic 25. Heroic LK was an incredibly difficult fight when it was current content, requiring intense coordination to beat (fuck those valks, fuck defiles, Shadow Traps were hilarious though). That was in ilvl 260-280 gear at lvl 80 though. The difference between a lvl 80 character in that ilvl vs a lvl 90 in 565 (like Az the youtuber is) is staggering. We're talking your average dps having like 30k hp in Wrath. Unbuffed, I have 713k hp right now according to my armory. That's more then 20x the hp.
The bosses that Az is teaching you how to solo are old content bosses, not current content. These are bosses that were hard in Vanilla, BC, and Wrath, but are jokes in MoP gear. The set of videos even clearly says "Old Bosses" in the title (https://www.youtube.com/playlist?list=PLWcyGtiFsXFXMSp7_67l8xHstBUJOU6SW).
http://eu.battle.net/wow/en/character/thunderhorn/Azbats/simple is his armory page. If you click on achievements on the left, then dungeons and raids, you can see his achievement date (the date he first gained said achievement) you can then narrow his dungeon/raid achievements down per expansion, and see the kill dates he has for various bosses across the entire game.
They are all very, very bad. Moreover, he doesn't HAVE the achievement for any of the heroic bosses this expansion, not even t14 heroic bosses that would be incredibly easy (as they are tuned to be beaten by people in 489/496 gear), which is an indicator that he doesn't really care about current raiding.
This is perfectly fine. I'm not judging him, nor am i judging you, for not doing this stuff.
However, it IS one of the two major metrics used to determine if someone is 'a good player' in terms of actual player skill, the other being pvp achievements. I can say that I'm a good raider because I've killed every boss when it was relevant since the start of wrath. I was getting US top 80-100 kills (which is awful, in retrospect) in wrath, US top 50-80 kills in Cata, and US top 5 kills in MoP. http://us.battle.net/wow/en/character/mal%27ganis/ikestorm/advanced is my main.
If you ever have Elemental Shaman questions, feel free to ask : ).