Alteration
Main purpose is protective, an alternative to armor for mages. It's not terrible at this, but it's also beaten pretty clearly by it's competitions as all significant penalties for armor are negated later on in the game while alteration spells will still cost a chunk of magicka and won't be on all the time like armor. Through perks, it also offers magical protection - 30% absorption and 30% resist magic is pretty nice. However, there's a large perk investment to reach those, and enchanting and other means of acquiring such seem like superior options. Paralyze is strong but Illusion offers superior crow control options. Overall a pretty bland school and one I'd stay away from personally since pretty much everything it does can be done better. It has one gem of a spell however, an obscure one that luckily doesn't require any investment into the tree. Equilibrium converts health into magicka at a decent rate. This is great for training restoration for obvious reasons, but also simply a nice tool to have as there are plenty of occasions when that magicka is more important than the health it costs.
Strength/Usefulness 4/10
Design/Fun Factor 3/10
Conjuration
Conjuration is essentially a magical alternative or supplement to companions. It serves this purpose very well, with powerful summons and thralls to take the heat of you as well as dish out damage. A certain summon is probably overpowered even, I think any conjurers will know which I'm thinking of. You can go heavily into the tree for improved atronachs or undead and eventually two of them, but even without heavy investment you get some solid options with just the expert level spells. It can also bring you temporary magical weapons which don't really compete with the best smith improved weapons, but still hold their own pretty well and are especially great early game with the perk to improve them. Unfortunately, the bound weapons are limited compared to previous games. Still, easily my favorite school in Skyrim.
Strength/Usefulness 10/10
Design/Fun Factor 9/10
Destruction
It's name makes it's purpose fairly obvious, but unfortunately it fails to deliver for many players. It's damage doesn't scale and it's high level spells cost too much magicka. Some players report a positive experience or at least have been able to make it work for them using various tricks and techniques or other skills to support it, but I'd recommend passing on it unless you're adamant about shocking, burning, and/or freezing stuff which is entirely understandable. Depressing, considering they've done so much with the actual spell mechanics in destruction - placing walls and rune traps in particular I was hopeful about. It does have a secondary purpose however, perhaps not intended to be as powerful as it is, the impact perk is probably the best tool in the game for keeping enemies off the player.
Strength/Usefulness 3/10
Design/Fun Factor 7/10
Illusion
Illusion is mainly about control, but also a stealth supplement quieting your spells, your footsteps, and even making yourself invisible. The control aspect of the tree requires some pretty heavy perk investments, but will end up being very powerful for escape or avoidance, or for making enemies do your work for you by inciting them into violence against each other - which is amusing if possibly overpowered. A good school but demands a lot of perks just to make it's spells reliably function at higher levels, and probably too powerful once you have.
Strength/Usefulness 9/10
Design/Fun Factor 7/10
Restoration
Maybe not the most exciting, but certainly one of the most universally useful. Most if it's best perks are very accessible in the lower part of it's tree/sign, and to do it's main job you don't need a heavy investment into the tree. It's non-healing spells however are pretty lackluster. Wards are a waste of mana, and the undead specific spells are, well, too specific to be worth spending perks to improve. Wards had potential to be an interesting mechanic to use in combat at least, but still a very lacking school. It's a one trick pony currently, though trick just happens to be one of the best. Unfortunately though, it's one trick is undeniably done better via potions, but I'm not sure I can dock restoration any points over this, since alchemy is obviously broken.
Strength/Usefulness 6/10
Design/Fun Factor 3/10
To mirror this very good post.
Alteration is not worth the investment. You need to perk it's tree deeply just to make the magicka cost of the spells worth it and ultimately a late game battlemage in smithed daedric gear with appropriate enchants is still going to be better than you at a cheaper cost.
As so many others have said Conjuration is one of the best if arguably the best school in the game. Solid perk tree a bound weapon that can easily be perked to be your soul binding weapon is(perhaps sadly) going to one of your strongest damage spells effectively for a long time and in terms of magicka cost outshines a great deal of Destruction. Some play styles might avoid using them though. And runes are crap for their damage to cost ratio.
People like to deny it but....Destruction is bad. It doesn't scale, it doesn't curve in advancement well. It's magicka efficiency is bad and required Alchemy/Enchantment investment to compete with the better trees.
I can't say anything more on Illusion that Hermit didn't.
Beyond that Wards are a crap I can't say much more on Resto. It's boring but it does what it needs to. The anti undead stuff is mostly useless but the healing recovers stamina perk and the 50% improvement perks are going to be highly useful to my battlemage once I'm in proper gear.