Basics of world soloing
Before we get into the flashy, fancy stuff, let's take a quick detour to talk about the fundamentals of soloing the various encounters that you find while out in the game world. Usually, these are things like group quests that pit you against a powerful elite monster that's usually a bit difficult to handle on your own. More often than not (well, depending on where you are, at least), it can be quite difficult to get real groups or even just anybody for the purpose of downing these fiends. Instead of giving up or wandering around seeking out someone to help you (note that asking for help is awesome even if you don't really need it, as someone else might need your help; just don't sit there doing nothing but spamming general until you find someone), it is always best to go out and attempt the quest on your own. When doing this, there are a few basic rules to keep in mind.
The rooting rule
Entangling Roots is the god of soloing spells. There is honestly nothing else out there that can even remotely compare to the power that Entangling Roots holds for allowing a player to single-handedly tackle a daunting task. When you find yourself face to face with an angry elite, the first thing you should always do it to test whether or not it is immune to Entangling Roots. If you can root it, you can kill it. With very few exceptions, a vast majority of quest elites are restricted to melee attacks or a smattering of ranged abilities that have a cooldown and are used infrequently. Very few are pure casters; any melee elite that you can root can be killed. Remember, there is no diminishing returns for CC effects in PvE, so you can keep the mob rooted indefinitely while you safely kill it.
If you are facing an rootable mob, there are two ways to go about killing it. The first, and my general favorite, is to merely attack it pretty much full out. Using Moonfire, Insect Swarm and Wrath/Starfire, you can just nuke the beast into oblivion. Obviously Entangling Roots is going to break quite frequently when following this method, making it slightly more dangerous, and you can do two different things in order to counteract roots breaking. First, you can alternate between Entangling Roots after every nuke or every other nuke to ensure that they stay up. If you are using this method and casting Wrath, then you need to stagger your cast of Entangling Roots or you'll just end up breaking it when Wrath actually hits the target, since it has a travel time while roots does not. The other way is to just sit there and let your DoTs do all the work. Doing this is much safer, since Entangling Roots will break less frequently, but it also takes longer.
The water rule
If Entangling Roots does not work (and even if it does), the second thing to test is whether or not the mob can be knocked back via Typhoon. For the most part, only the larger of mobs are immune to Typhoon's effect, although this isn't always the case. Typhoon will not save you from taking several hits from the mob, but it does give you something precious: time. Beyond just the knockback, also check to see if the mob is being slowed by Typhoon or not. Having it do so opens up extra options for how you can make use of Typhoon's effects.
The key to using Typhoon, as I said, is that it buys you time -- time to get a few quick HoTs up and be back into Moonkin Form before the mob is back in your face. Healing is a very important part of soloing mobs, and you always want to be avoid taking as much damage while out of Moonkin Form as you possibly can. If you have a good connection and a clear path, then using Travel Form to kite away for a bit will buy you some additional space in order to heal with, but be careful not to run too far or you'll just end up leashing the mob.
When healing, always open up with Regrowth, assuming that you aren't being hit on; then get a Rejuvenation up. If you have more time after then, toss out a Lifebloom as well before shifting back into Moonkin Form. If you still have a mob on you while attempting to heal, then go in reverse order; get up Lifebloom and Rejuvenation early and often and use Nourish instead of Regrowth if it's an emergency. Always try to limit the amount of time any mob is attacking you outside of Moonkin Form.
The forest rule
There is one more trick that you can use as a balance druid to try and assist with soloing some of the tougher monsters out there. Force of Nature is a very powerful cooldown, with the Treants that it summons dealing a hefty amount of damage. While it is nice to use this in a burst tactic against an elite to try and take it out before it gets you, there is another method of using them. Open up against the mob with Faerie Fire just to get initial aggro, then instantly summon your natural fury against them. Usually, before the mob even gets close enough to land a single swing, the Treants will pull aggro off of you and begin to tank the mob. During this time, very carefully use Insect Swarm and Moonfire to help whittle the mob's health away without pulling aggro onto yourself.
Although the Treants from Force of Nature are not the best of tanks, they do have some amount of health, and by the time the mob has gone through killing all three of them, they've generally spent a significant amount of time dealing with something that isn't you while their health was slowly drained away. The trick here is to be excessively careful with threat to ensure that the Treants remain as the focus instead of you. Normally, you can get away with using a few Wraths after a couple of seconds, but don't outright spam the spell; pace yourself, and allow your pets to do their thing. If the mob is able to be effected by Typhoon, then make sure to use it when your last Treant dies in order to give yourself as much time as possible before you start taking damage. This may seem silly, given that the Treants will most likely die well before their natural 30 seconds is up and leaving them dealing less damage than if you had been the tank, but it saves you all of your other cooldowns, mana from healing and buys you a significant amount of time.