Assets represent a character's special qualities that aren't covered by any Traits, plus a character's significant possessions, privileges, and other benefits. This is where an imaginative player can have a ball really customizing his or her character.
You have 10 dice to distribute to your Assets, max of 5 dice to any one Asset. Wanna be strong? Make up an Asset like Steel Cord Thews and assign dice to it. Wanna be Dejah Thoris? Make up an Asset like Incomparable Beauty and assign dice to it. Wanna own a neat suit of armor? Make it up and assign dice to it.
My setting books for Vivid will include suggestions on what unique Assets characters can take, divided by region/culture. As a GM I'll normally enforce restrictions on what a character can get based on their origins - for example a player in Sea Rovers of Syrene who wants a Saberhorn Bow will have to be a Palmarian or Felyran, this item only being available in Palmaria.
The most important function of Assets is to aid in actions by providing bonus dice. You may 'tap' an Asset for extra dice to roll until it is exhausted, after which you must take some kind of action to refresh that Asset. For example, a Sorceress with the Asset: Beloved of the Wind Spirits 4 can get up to 4 extra dice when casting wind spells.
Another very important function of Assets: damage is soaked or avoided by spending dice from your appropriate Assets. Armor and shields are of course Assets, but again here a creative player can come up with ways to use other Assets to save their characters.