Action bars are the section(s) of the screen where abilities can be placed in order to be used quickly. By default, there is one action bar on the screen, along the left half of the bottom of the screen, whose buttons correspond to the keys along the top row of the keyboard. (1 thru =.)
When a character is first created, a few basic abilities are placed in the action bar, including basic attack, two class abilities, at least one racial ability, and the Food and Drink in their backpack. Pressing the key that corresponds to the box that item is in or clicking that box with the mouse will activate it.
As your character gains new things to put in an action bar, they will not appear there immediately. You will need to open your spellbook (default key: P) and find that ability in it. Then drag the ability onto the action bar in the position you want it. If you replace an old skill, it will then be put on the cursor and you can either relocate it or click in the main window to "drop" it.
There are things that can be put on an action bar besides class abilities, however. For example, food and drink, or any other item that is used by right-clicking it. To place these items on your action bar, find them in your inventory (or your character window) and drag them to the action bar the same way. When clicking on the action bar, use a left click even though you would normally right-click these items from a bag.
The final option for adding to an action bar is a macro.
There are two ways to give yourself more places to put your activated abilities. The first is to simply cycle to another bar. At the right end of your action bar should be a number one, with a small up and down arrow above and below it. If you click these arrows, or press shift-up and shift-down on the keyboard, you will move to a fresh action bar. You have six total bars.
However, this isn't very convenient. To place an additional action bar on the screen, you can open the options menu by pressing Escape, click on Interface Options, and then go to the Advanced Options tab. At the top of the screen will be choices to add other action bars -- to the lower-left of the screen, to the lower-right, and two down the right edge of the screen. If you choose to do this, the action bars will be clear, so you may not realize they are there until you try to drag an ability; the boxes darken so you can see them when you do this. You can not set hotkeys to these extra bars.
It should also be noted that these are actually the same six bars you can cycle between, and each position "takes up" a particular bar, meaning you will no longer cycle along that bar. Because of this, if you were using the cycle method first, you might have some items in your new bar when you first add it.
Some classes have certain modes they can enter where they would use different ability sets. Rogues, for instance, have certain abilities that only function in stealth. Druids and Priests have at least one form that typically uses skills as well, and Warrior stances all have restrictions on what can be used with them. In each of these cases, if you enter one of these alternative modes, your first action bar (#1) is replaced by an alternate bar. You will need to fill this bar with the skills you want on it, but the game will automatically switch to this set when you, for example, enter Bear Form.
Hunters and Warlocks will also find that their pets will have a special bar as well, that appears above the main action bar. This smaller version has the essential tools for instructing a pet on basic ideas like "Stay" or "Defensive"ly fight back only.
Many of the most popular mods are simply to rearrange where things can be found on the screen, and one of the primary pieces of the screen that often gets shifted around are the action bars. For players that prefer to click on their abilities with the mouse, it helps to have them more centralized, or have better access to more of them. Many mods make a solid block out of the abilities, or reduce the size of them to fit more on the screen. If you find yourself frustrated with the layout of the Default UI, you may want to check out some of the add-ons available to put things in more convenient places for you.