As a DM:
1. XP are a measure of experience - how much a character has grown in knowledge, skill, power, and ability. It is a tool that allows me to pace the adventure, series, or campaign more accurately to the PCs' capabilities.
2. XP are a reward - for characters that have stretched their limits and for players who like seeing their characters (and NPCs) grow and change.
3. XP are a resource for casting spells and magic items. (I have real game mechanics problems with this: A wizard becomes *less* experienced after he makes a magical item?!?!)
As a Player:
1. XP are a reward that serves to increase my character's power, allowing me to add new dimensions and elements to the character. I have, in my mind's eye, an "idealized" vision of my character - the goal that the character works towards that may include multi-classing, prestige classes, or other mechanics-related elements. XP are the progress meter towards that goal. Note that this is not the only facet of that image. Depending on the character, land-ownership, revenge, military/political/magical power, contacts, networking, and so forth all factor in as well.
2. XP represents level-advancement that allows my character to take on bigger and tougher challenges. These challenges may border on the impossible - but they only border on it after XP awards. They're not *quite* impossible. When my character is low-level, I thirst for the time when I can take on giants. When my character is mid-level, I thirst for the time when I can take on dragons. When my character is high-level, I thirst for the time when I can take on extra-planar threats. When/if my character is epic level, all bets are off
3. XP represents a resource. It gets spent on permanent negative levels, casting certain spells, and creating magic items. (I've already mentioned why this irks me as a DM.)
4. It isn't all about power-tripping either. Depending on the game, players, and DM, XP also represents the increase and improvement in skills and abilities that grant greater role-playing range. I also thirst for the time when my bard, horribly scarred by the black pudding's acid, sings so sweetly that the entire Imperial Court is moved to stunned, silent tears at the pure, ephemeral beauty he evokes, while hiding his hideous visage behind a cloth mask.
I want my characters to be ~LEGENDS~ in the campaign, not just showered in glory for feats of arms. XP represents a progress meter to achieving that goal.