The personal story:
In Guild Wars 2, a lot of attention has gone into creating a good story. Few MMORPGs have a story on par with those of singleplayer RPGs, and ArenaNet are trying to do something about that.
When you create a character in most modern MMOs, you're asked to provide appearances, race, class and name. Your "character" has no background beyond the one in your imagination.
In Guild Wars 2, you will be asked questions to fill out some background information for your character. Where are you from, what's your social background, stuff like that. These will help define your character, your purpose in the world, how people react to you and your personal story as you play through it.
Every character in Guild Wars 2 will have their own unique home. This is a small instance in your racial capital, that changes depending on the choices you make during the game. As you make new allies and lose old, join orders, and make other meaningful choices, you will see consequnces in your home instance, whether these are good or bad.
You can invite players to your home, but you can only see other people's home by invitation.
A lot of effort has gone into making the personal story in Guild Wars 2 truly personal. It will be shaped not just by the choices you make when you create your character, but by the choices you make throughout the storyline as well. You can therefor create several different characters who can experience very different stories.
Currently, there are thousands of different story combinations to explore.
Choice is not just about morale though, as many missions will offer tactical choice as well. Do you attack first thing at dawn, or do you sneak in under cover of nighfall?
Playing the game:
You can invite friends along to help out with the tasks in your personal story, but your friends' stories will not be affected by this. All the storycontent is also designed to be solofriendly, so you can go at it alone if you so choose.
There will not be any material rewards that favor one choice over the other. So you'll never be faced with a choice, where there is a level 1337 "exploding firesword of fiery burning" on one end and a level 2 "splashy fish of water and digestion" on the other.
Your personal story will also remain seperate from everything else, so no matter what happens in the dynamic events or whatever else can happen, your story will remain unaffected.
And while you still can't recieve quests like you do in most MMOs these days (see the piece on the dynamic event system), you can recieve visits in your home instance from NPCs who would like your help. There will still not be any exclamation marks, but they will actually walk up to you and ask for your help.
Personality refers to the behaviour of your character. As you behave one way or another towards NPCs in the Guild Wars world, you'll be seen as a personality archetype. Maybe you're a badass character, who would rather threaten their way to help or goods, or maybe you're the gentle character who would rather inspire the NPCs to get the same results. The NPCs will then react according to the archetype you have about you.
This is mostly cosmetic though, and its main purpose is to allow for greater immersion as players will have an easier time roleplaying and getting into their character.
There will be rewards for certain interactions with the right personality, but these rewards can be obtained other ways as well, through buying for example.