Strengths: Warriors have a number of abilities for dealing damage (duh, they're warriors, right?), and they can use any weapon under the sun one-handed or two handed. Specced correctly they are capable of respectable melee damage and their plate armor gives them strong defense. They also have unique abilities to close distance and finish fights quickly.
Weaknesses: Consumable dependant. It is very easy to get overwealmed because although warriors can deal damage effectively to multiple mobs, they can't self heal so it becomes a battle of attrition. A poorly prepared warrior will eventually lose, repeatedly. Potions, bandages and food are must haves. Another facet of this is downtime. Warriors have a period after fights where they have to eat, bandage, etc and this leads to frequent, substantial downtime.
Strengths: Depends on spec. They are premium tanks with many varied abilities, strong aggro management perfectly suited for main or off-tanking.
Weaknesses: DPS warriors are not a strong choice for groups being inferior to deathknights and rogues in the melee DPS category. They lack crowd control abilties in general, although they have some abilities to stop fleeing mobs and casters, they are comparatively weak in this area. In summary, if you have a non-tank warrior in the group, they're a weakness. Protection warriors suffer from the sole weakness of having to build up rage to generate threat, so fights have to open slightly slower to allowe them to build up threat.
Strengths: Depends on spec. DPS warriors can generate excellent burst damage. They also have good intercepts and abilities to break fear. They have tools at their disposal to counter casters and melee opponents, they are a strong PvP class.
Weaknesses: Although they do have good abilities to close distance, their abilities lack range as a whole and their damage dealing is melee based so they are generally ineffective at range. Also, although they have great capabilities for burst damage, much of this relies on rage which is not generally available immediately. Protection warriors, although tough to kill, are a PvE spec and should not be used in PvP.
Protection: This is the tanking tree. It is one of the most useful and well rounded tanking trees in the game. It offers solid mitigation, a multitude of options for pulling aggro, and due to recent patches, usefulness in AOE situations. Protections warriors are extremely useful as either main or off tanks due to their mechanics.
Fury: This warrior tree revolves around critical strikes, both giving and receiving. It has abilities to regain health and increase damage. This is also mainly the tree which specializes in dual wielding. The ending talent in this tree allows the use of a two handed weapon in each hand.
Arms: This tree has numerous abilites generally involving the use of a single two-handed weapon. It allows easier and more efficient use of some of the warriors' main countering abilities and is good against solo or group mobs.
Warriors are really a mixed bag. It is possible to get solid DPS from them at the expense of downtime, although the DPS they put out is generally not as solid as the caster classes, and since they're taking damage generally, they are requiring more down time. I can't reccomend a warrior for leveling unless you want to tank. DPS warriors are frustrating to level and even more frustrating to try to find groups with. Bottom line, they make some of the best tanks in the game, always have since Wow was released. They are a passable option in PvP. If you are looking for more than that, look elsewhere.