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The spellbook is a panel, accessible through a hotkey (default key: P) or the icon located in the bottom right corner of the screen, which holds all of your character's abilities. It is divided into 4 tabs - the 'General' tab, and 3 tabs that represent the three ability trees of your class.
Found in the General tab are any Racial Bonuses you may have, as well as racial abilities that activate on use. Also in this tab are your professions and any abilities that arise from them, such as Find Herbs or Smelting. Passive abilities such as Dodge and Parry are in the General tab, and the auto-attack icon.
To move an icon from your spellbook to an action bar, simply click and drag the icon into its desired place on the bar. If you wish to replace an ability that is on the bar already, drag it onto that icon and either relocate or delete the old ability. (To delete it, drag it from the actionbar into an 'empty' space on the screen, such as over your character.) The other 3 tabs depend on your class, and may not even show up until you train in an ability of that type. The highest rank of each ability will appear.
Abilities can also be used directly out of the spellbook by clicking on them. To add a spell to a macro, open your spellbook while editing the macro, and shift-click the spell.