Experience is a numerical value that is used to determine when you reach the next character level, also known as "leveling up." Every newly created character starts at level 1 with no experience. Each time you, or a member of your party (whether it be your hireling, your minion(s), or another player) kills a monster, you gain a certain number of experience points. There are two places where you can view your current amount of experience. One place is at the top of the Character Screen, and the other is the Experience Bar found on the user interface at the bottom of the screen (see image below).
When your character reaches the amount of experience needed for the next level, he/she will "level up." The moment this happens, you will hear a special sound, and the gray cross buttons on the user interface will become red. 
Each time you gain a character level, you receive 5 stat points to spend on attributes, as well as 1 skill point that you can use to acquire a new skill or upgrade a skill you already have. On the user interface, clicking on the left red button opens your Character Screen, while clicking the right button opens your Skill Trees.