Level downscaling will downlevel your character to the maximum allowable content level for that area.
So this will affect all your primary stats, Toughness, Vitality, Power and Precision. When you open your hero panel you'll see your main stats on the right in white, and next to them on the left you'll see your 'effective' stats in green. These are the numbers that you're going to be using. As these numbers are changed this will effect your damage, hp, crit chance, etc.
The thing with you feeling more powerful is that you still have the higher level gear. So for example, if you are lv30 you might have an amulet with +11 power. You go to a lv10 area, and fight with players who at lv10 can't even equip an amulet. Now I'm assuming, but let's say that the downscaling is a %, then your overall stats will be higher, meaning you'll be stronger at lower level because of your better gear.
You are by no means at all invincible, but you'll certainly fare a little better due to having more or better gear. The real killer is that as a higher level you will have traits and skills available which lower levels will not. You can also use your Elite skill still, if you've unlocked it. So there are some advantages to revisiting the lower level areas.
I think most people's experiences of this will be doing each of the race's starter areas once they've finished their own.
You can read more on the wiki, http://wiki.guildwars2.com/wiki/Dynamic_level_adjustment
Character level adjustments have a limit, however. The
level-adjustment system won’t let a level 1 character go head-to-head
against a level 80 character with much chance of victory—a level 80
character is still going to have more bonuses on their gear and access
to a full complement of utility and elite skills that players just
don’t have at lower levels. This is not to say that the high-level
character will one-shot the level 1 player, but they will have a clear
advantage.
http://www.arena.net/blog/mike-ferguson-on-guild-wars-2-world-vs-world
Edit
Since Jan 28th 2013, the downlevelling has changed. https://www.guildwars2.com/en-gb/news/dynamic-leveling-adjustment/
Take a look at the Wiki on the page Mystic Forge. It will tell you where the forge is, how it is used, and some predictable recipes. The forge lets you do two things. First you can create recipes to craft other items like consumables, weapons, or armor. Second, the forge can be used to create better crafting materials of one type higher.
The examples the wiki gives are as follows:
Common Crafting Material Promotion
Common crafting materials can be promoted to a small pile of higher
tier materials with the following formula. The tier of dust needed is equivalent to the tier of the wanted product. So, if you are hoping to create Orichalcum, a tier 6 common material, you need to use Crystalline Dust, a tier 6 dust. If common crafting material B is one tier higher than common crafting material A, then:
10-50 Crafting Material B = 250 Common Crafting Material A (i.e. 250 Mithril Ores) + 1 Common Crafting Material B (i.e. 1 Orichalcum Ores) + 2-5 Dust A (i.e. 5 Pile of Crystalline Dust) + 5 Philosopher's stone
In that example: That recipe will produce 10-50 Orichalcum Ores
Armor boxes and armor satchels
Recipes can be created using the Mystic Forge. They will allow you to
craft armor boxes and armor satchels, which will in turn give you
every piece of an armor set.
Armor set recipe
Ingredients
5 Bottle of Elonian Wine
20 Arcane Crystal
50 Mystic Coin
1 Armor item
Mystic coins seem to be a common item needed for Food Trays, Weapons and Armor. Also look at this page for current recipe research.
You can see this in action with this video on how to make a Giant Chocolate Cake. Details are in the video summary.
Best Answer
EDIT Original information removed as it's now out of date and inaccurate.
EDIT: As per january 28th, the developers made some changes in the way level downscaling works. According to the article, the net effect is to make downscaling both more challenging /and/ more rewarding.