Balance is achieved via a process of iteration.
For initial balance a rough guess suffices. You use your head and experience of other maps to come up with stats that sound like they could be balanced. If you want a unit to be strong then it should have appropriate costs to enforce rarity. If you want a unit to be weak then it should have appropriate costs to enforce commonness.
However it is almost impossible for the values you choose to be truly balanced. From here on it becomes a process of iteration.
1. Analyse balance.
2. Identify possible changes.
3. Implement changes.
4. Go back to 1.
The biggest problem is analysing balance (step 1). How do you tell if something is balanced? There is no really good solution to this as I will outline below.
Many inexperienced developers would ask players for possible balance change suggestions. The logic is that if a lot of people agree something is too strong/weak then is must be so. This is flawed because although it will be right for major imbalances, minor imbalances might be overlooked or even made worse. An element deemed "weak" might actually be very strong if used properly by an experienced player and so buffing it would result in it becoming too strong. Players also tend to not know what they want so a lot of suggestions they give have no evidence to back them or are the product of a selfish dream.
Another approach is to analyse play data. The logic here is if an element is noticeably performing out of the intended performance average then it has to be unbalanced and need adjusting. This works great for some kinds of balance such as PvM when trying to stop underperforming classes but does not actually result in true balance. Cultural bias might influence choices resulting in uneven uses of functionality. Some elements might need hard to obtain playing skills to be used effectively meaning that it under performs most of the time but over performs other times which will not be detected with this type of analysis. People might also troll your data by forgery (losing on purpose, doing stupid things, messing around etc). As such this type of analysis will not allow you to pick up small balance problems.
The only real way to balance something is by asking "experts". An expert is someone who has a high level of player skill and under stands the content provided fully. They usually play the game a lot and will outperform the most people. Due to this knowledge they can identify more accurately where they struggle which in turn might become a possible target for a change. You can combine the above two techniques to experts for the best results. They provide much better balance ideas than most players so you can often map their suggests direction to changes. They also will provide much better data in their play response as they will use the game resources more efficiently. This is the technique used by real game developers such as Blizzard to balance StarCraft II and Diablo III.
A big problem is who is an expert? If you make a game and play it a lot then you might become an expert (like the SC2 developers) but you do not necessarily (if you seldom play your own map in WC3). Also just because someone has played your map a lot does not make him an expert as they might not fully understand it despite the amount they played. In the case of WarCraft III you can have experts who barely play your map at all due to the similarities with other maps. In the end there is no real way to tell who is an expert and who is not, you will need to decide that for yourself by trusting your instincts.
I will conclude with some overlooked balance aspects.
These abuse the standard combat AI of WC3.
Kiting:
In many PvM maps you will eventually have to face something powerful and tough. A newbie will run up and "face tank" it which can easily result in him losing his character and the entire team dying due to no tank. An expert might grab its attention and just keep running from it. Even if it is faster than him he will take far fewer hits and the rest of the team is free to stand attacking. This strategy can change week tanks into very strong tanks, especially if they have a skill like taunt.
Ping-Pong:
If 2 range units face a strong enemy that out speeds them they will usually die. An expert might use them to execute a ping-pong by placing them at opposite sides of the strong unit and get each range unit to grab the attention of the strong unit in turn. The unit will thus remain relatively stationary as it walks back and forth between targets. Result is the strong unit dies without every landing a shot on the weak unit despite being faster and stronger.
Strafing:
A range unit will generally die to strong enemies in open combat without someone to tank. An expert might build the range unit's movement speed up to high levels so it out speeds that of the strong enemy. Although the range unit will still die if forced into combat he does not ever need to enter combat. The expert can keep the range unit back out of harms way and shoot freely at incoming enemies. When the enemies are too close he can run further away and repeat. Especially if the unit has a lot of range this becomes easy to do and even works on other range units.
1 man stand:
When confronted by a huge inescapable mob many newbie players will stand ground and die fighting while the mob surrounds them. An expert might retreat to a corner or even a wall to put his back against. The result is that the mob will be less effective as it will be able to dish out less damage. Especially if a choke point is used then instead of 7 melee units hitting him from all sides he may only need to face them 1 on 1 turning a hopeless situation into a guaranteed victory. It even works on range units to some extent as you can reduce the number of units that are in range to shoot at you. This also applies to groups of units since it will stop your units being surrounded when hopelessly out numbered.
Ranged Push
Many newbie players with an army of ranged units will carelessly attack move them into an enemy position. The result is that the units will stop as soon as they come into range of the enemy blocking the way for other units. Especially if the enemy has longer range than you this will give them free hits and allow them to kill your range units before they have a chance to move into range. You can also run out of circumference area of a target resulting in range firepower being wasted as units are blocked out of range by your own firing units. An expert will normal move the range units close to the target before attacking. This will potentially allow all the range units to attack very quickly giving them full firepower. Especially when numbers are on your side but superior range is not as this will greatly improve range unit combat performance. Obviously you never use this on similar range mobs to your own as you will only sacrifice shots to give them free shots.
These are but a few that can drastically alter the balance of a unit. Some can even turn a "weak" unit into something that is unstoppable such as strafing.
