Playing a Paladin is in many ways a balancing act. Sure you can heal, tank or DPS, but you can't do any of them especially well unless you specialize in one of them through gear and talents. By doing so you give up much of your flexibility due to specialization and spending time on that one area, this leads to becoming weaker in the other two areas. Part of playing a Paladin is finding the balance point that suits your play style. Do you want to specialize in one thing or stay a jack of all trades?
As mentioned above, playing a Paladin involves a lot of choices. As soon as you get to level 10 and are able to select a specialization you will have to start making those choices. From that point on you should start focusing on your style and collecting gear that will enhance it. Between your specialization, talents and gear you can be formidable at whatever you choose to do. Be prepared though for those times when you want to play in a group and they need something other than your primary role.
For those times you need to do something else, make sure you have a collection of gear either in your bank or bags that allow you to fulfil other roles. If you are mainly a DPS based Paladin for example, make sure you have a set of healing gear for those times when you might need it. Doing this will allow you to join far more groups that you could normally find if you just had gear for one thing. Even though you are not specialized in what that group is looking for, with the right gear you can be competent.
There are few other classes that can take as much of a beating as a Paladin and none that can do it as well on their own. Sure a Warrior, Death Knight or Druid in bear form can have as much or more armour and health, but neither can heal themselves (unless the druid drops to normal form and takes a beating). As soon as you hit level 40 you are the only class other than Warriors and Death Knights that can wear plate armour.
Between good armour, a shield, stuns, heals and your bubbles you can survive or escape almost anything within several levels of you.
When you get in a tight situation focus on mana and health as a priority and don't worry about doing a ton of damage. Wear the opponent down over time. this should include having a seal of light or wisdom judged on them and the opposite up on you. Use one of your bubbles or stuns whenever you can and bandage yourself during that time to conserve mana. Drink mana potions instead of healing potions as you can heal yourself with mana.
Run as a last resort. However if you use the above method you can generally survive an enemy several levels higher than you are quite easily. The fight will not be quick (sometimes 5-10 minutes) but you will survive.
When playing the game solo, you really only have 2 options. Play as Retribution or play as Protection, even then Protection builds are not really viable until at least level 40 for soloing. In most cases you will want to solo and play as a Retribution Paladin so that you can cause as much damage as possible, get out of combat, heal, and look for another fight. If you ever really get into trouble you have stuns, bubbles, and heals going for you.
Your role in a group as a Paladin really depends on what specialization you are. Because each one is so different we will look at them one at a time. Before we do though realize that all Paladins should ensure they have the most beneficial blessing up on the group, factoring in themselves and other players in the group. Also make sure they do not overlap with any other Paladin in the group.
Protection Paladin - When you spec into Protection you are becoming a tank. Your role in the group is to get things mad at you, keep them attacking you, and taking as much of the damage as you can. One of your best tools as a tank is your Righteous Fury ability which lets you generate extra threat from all of your holy damage caused. In addition to this you have Consecration, which will cause holy damage to everything around you. Team that up with Holy Wrath that hits everything nearby with additional Holy damage, Hammer of the Righteous that deals holy damage to nearby enemies, and Shield of the Righteous that bounces between enemies, and you have a solid grasp on things. Between these things Paladins have gained the reputation as the best AoE tank in the game.
Check out the updated: Big Guide of Tanking Mists of Pandaria Edition for lots of great tanking information.
Outdated but still usefull - There is a lot more to being a Paladin Tank in a group though than just keeping Righteous Fury up and casting Consecration. There is so much more that we have created a separate guide on How to be a Tankadin and discussed it more in Cataclysm Shakes Up Paladin Tanking Â Semi-Hardcore #4.
Retribution Paladin - This type of Paladin is the damage dealing melee monster. No other Paladin build can do as much damage in most situations as a Retribution Paladin. Retribution Paladins can do significant AOE damage while mixing it up in melee by using their Seal of Command talent, Divine Storm, Holy Wrath and Consecration.
There is a lot to know if you chose to be a Retribution Paladin and smite your foes with your righteous powers. There is so much more that you will need to know that we have created a separate guide on How to Be a Retadin.
Holy Paladin - As a Holy Paladin your job in a group is probably the hardest. Your job is to keep everyone else alive. Primarily that means your efforts will be focused on the Tank in the group, however with all the DPS players that like to jump the gun, or Tanks that can not hold aggro on more than one target at a time, your life is likely to be hectic healing one person and then another. One of your most important talents as a Holy Paladin is your huge heal called Divine Light, since it heals such a large amount. Also Your Holy Radiance ability which, while expensive mana wise, heals everyone nearby over time. You also have a self preservation bubble and a bubble that can protect another player or yourself. Other than that you have single target heals, but some of the biggest, fastest and most mana efficient in the game.
Being a Holy Paladin is complicated and much more needs to be explained than can be done in a few paragraphs here. To help you we have created a more in depth guide on How to be a Healadin.
This is another very often overlooked skill that truly separates the good players from the herd. Many normal players get focused just on what they are doing and lose track of what else is going on. As a multi-purpose class you can not afford to do that, if you want to be great as a group PvE player. There are times you need to be able to switch from what you are doing to do something else. For example, even though you may be in a group as DPS, realizing a healer went down, and dropping out of the attack to heal (even though they will be small) could make all the difference. Being a tank and knowing when to bubble another player (especially a healer) can make all the difference as well. Recognizing the times that you should switch roles and acting them takes practice and can make all the difference in a group.
For longer changes it may mean carrying gear for different purposes and being skilled at using it. For instance many Paladin tanks carry full healing or DPS sets for raids, so that when they are at a fight that requires less tanks than they have in the group they can switch gear and help out in another capacity. For these longer term switches you should have a secondary talent build ready to go.