Unless stated otherwise, all information herein is taken from The World of Robert Jordan's "The Wheel of Time".
Also called Gaidin (GYE-deen)
Warders are the companions of Aes Sedai, bonded to them in some way that others do not understand. Their loyalty is to their Aes Sedai above and beyond everyone and everything else. Most Ajahs can bond just one, but Greens are allowed as many as they wish and Reds may not bond any at all.
Traditionally, Warders have been men, bonded to a single Aes Sedai. Since the time of the Trolloc Wars, it has been a requirement that the man gives permission, though this is not always followed. Doing so, or bonding as an Accepted are serious crimes, though not quite stilling offences (TFoH, Ch. 35; TDR, Ch. 39). However, if discovered, the perpetrator would likely have to hand their bond on to someone else. Despite the tradition, it is possible for more than one woman to bond a man, even one who cannot channel if the weaves are laid for her, or to bond a man who is already taken (WH, Ch. 12). It is also known that a woman can be bonded in the same way as a man (TFoH, Ch. 35), though the effect of this bond is different and the two will often mirror each other's moods and even lack of sobriety (ACoS, Ch. 21). The prohibition the Reds have on bonding has also been rethought in light of the presence of Asha'man and the perceived need to control them (KoD, Prologue), and there have been plans to alter the bond so that the women do not have to suffer if the man goes mad (CoT, Ch. 30).
In general, the bond lasts until the death of one of the partners. As Aes Sedai live much longer than Gaidin, it is likely to be the man who dies first, but this is not always the case. Death affects each differently. Women are overwhelmed with a feeling of grief, for example when Leane's warder died the pain was so much she initially did not think she could ever bond another (LoC, Ch. 30). Men can simply die from the shock, or if not, temporarily go mad in an attempt to get revenge (This passage needs a reference). Even those who do not die immediately often die soon. They can be saved, by giving them a task important enough that they focus on it rather than their reckless desire to push themselves, but it is not easy (ACoS, Ch. 12)
The Warder bond is a thing composed of spirit, a complex weave of over a hundred threads, laid over he Aes Sedai, then connected to the man to be bonded (WH, Ch. 12), that was unknown in the Age of Legends (TPoD, Ch. 2) and discovered some time after. It is in some ways similar to the first sister bond the Aiel use (WH, Prologue), but with subtle differences in the awareness of the other person.
The bond gives the Warder the gift of endurance, quick healing and the ability to sense the taint on the Dark One or Shadowspawn. It also allows him to sense his Aes Sedai and an idea of where she is. The Aes Sedai can also sense her Warder and tell if he is approaching. The other gifts she gains are a closely guarded secret, but they are able to take strength from their Warder, even to the point of death and to use the Power to Compel them into an action (LoC, Ch. 52). This does not work on a man holding the Power (LoC, Ch. 10).
Warder Aes Sedai Pairs
The Asha'man bond
The Asha'man also have their own version of the bond, devised by a man named Canler, so that a man could sense his wife and know she was safe (ACoS, Ch. 27). This bond has been modified so that the control element is automatic. The Aes Sedai under the command of Toveine Gazal that were sent to destroy the Black Tower were bonded in this way and must obey any command their Asha'man gives them (TPoD, Ch. 26).