A few things to know about character experience points and how heroes "level-up" within the game.
As heroes defeat monsters, make good decisions and are otherwise rewarded by the gamemaster, they receive experience points. All characters begin their adventuring careers with zero experience points, and, as they accrue these points over time, eventually "level-up" and gain additional power and status. Players should always look for opportunities to gain additional experience points, especially outside of combat (again, the gamemaster will often reward good decision-making with bonus experience points, not just from the slaying of encountered monsters).
When a character reaches enough experience points to gain a new level (see the Classes page for specific amounts of experience needed to ascend to a particular level), that hero will automatically gain the new level--the player does not have to do anything special. Upon leveling-up, the character will automatically receive at least one hit point (normally more, depending on class, level and character Constitution) and the hero will then begin his journey toward the next level.
For magic-using classes (clerics and magic-users), note that when a hero levels-up, they also receive a new spell slot. At that point, the player must click the MANAGE SPELLS button and manually assign an appropriate spell to that slot. Don't forget to do this! Without manually selecting a spell for the new slot, the character won't be able to maximize her new powers having gained the new level.