Priests are the main healing class on WoW, but these tips apply to any healers. The biggest and most important thing a priest must do is figure out who needs heals and when. You need to use your mana in a conservative way, but still keep your group alive. You want to focus heals on people who are taking damage, and those who are low on HP.
When healing, it is best to watch the health bars on the top left of your screen. You will see the name and hp of each party member displayed there. When you want to heal someone, just click their picture and use the hotkey for your heal. This way you don't have to try to select them yourself, which will often end up grabbing the wrong player or an enemy. Keep a constant eye on their HP displays. You can also use the F1-F5 keys to select the party members, but this requires you to remember which number each person in the group is.
Another method I use is target of target. You have to enable a setting in your interface menu to get this to work. When this is enabled, any enemy will show the picture of its target under it. If there is just one enemy doing damage, you can click on the enemy and watch the HP of his targets right from his HP bar. Press the hotkey for a heal to ready it, then click on the bar displayed for his target to immediately start a heal. You can switch between targeting the enemy or his target by pressing the "T" button.