The Vigil of Stendarr is an order of holy warriors who fight in the name of the divine Stendarr. Vigilants can be found journeying across Tamriel in their dedication to wiping out Daedra and any other "abominations" (such as vampires, werewolves, and witches). The Vigil's base of operations in Skyrim is the Hall of the Vigilant, located south of Dawnstar. They also gather near Stendarr's Beacon in The Rift. The Vigilants in Skyrim are led by Keeper Carcette.
The Vigil has close ties to the Dawnguard in Skyrim; not only do the two factions have a similar focus, but the newly-reformed Dawnguard was started by two former members of the Vigil, Isran and Celann.
Vigilants can often be found walking in groups along roads, or found individually in certain places. Any Vigilant that is not in combat will cure any diseases you have, if asked. If you have lycanthropy, they will not attack you unless you approach them in beast form.
You may have to fight them during Daedric quests; for instance, when following Barbas at the beginning of the quest A Daedra's Best Friend, you might cross paths with a group of Vigilants of Stendarr who will attack Barbas and will have to be killed before you can proceed.
Vigilants of Stendarr make use of both one-handed melee attacks and defensive magic in battle. They are armed with a leveled mace (any quality from iron to Daedric), and all members have the Oakflesh, Healing, and Lesser Ward spells.
They wear the following outfit, which varies slightly depending on their level:
Charmed Vigilants are found in Ruunvald during the quest Bolstering the Ranks. A warlock named Minorne charmed a temple leader, Moric Sidrey, while he was excavating the Ruunvald site. Under her guidance, he gathered more Vigilants there, and they were all eventually charmed by her as well. They worship her as a god, and will fight to protect her against anyone who comes along. They die after Minorne is killed.
When you meet a Vigilant of Stendarr, you can ask a few questions about their purpose and their beliefs:
You're a Vigilant of Stendarr?
"Yes. Our order was founded after the Oblivion Crisis. We dedicate our lives to facing the threat of Daedra wherever they appear. We're quartered in the Hall of the Vigilant. Keeper Carcette heads Skyrim's branch of our Order there, providing healing and justice as needed."
Who's Stendarr?
"He is the God of Mercy. The patron of order and justice for all of Tamriel. We bring his compassion where none can be found, by cleansing all those who would offend his children."
You hunt daedra?
"And any other abominations that prey on mortals. Vampires. Werewolves. Witches. But the Daedra are the worst. Their callous disregard for our lives is abhorrent in the eyes of the God of Mercy."
What's dangerous about daedra?
"They are profane creatures with no hope of redemption. They lure innocents to their deaths and corrupt the souls of all they touch. And their heretical artifacts grant powers to mortals that wreak havoc across Tamriel. They must be destroyed or guarded to prevent temptation."
If you have a disease, you will have the option to ask for healing:
Be careful when using Healing Hands; if a Vigilant of Stendarr is healed by this spell while out on the road, there is a chance they will become hostile. They will also turn hostile if under the effects of Rally.
If you accidentally strike a Vigilant of Stendarr while in combat, there's a chance that all Vigilants will turn hostile.
They can sometimes be found carrying Daedric weapons, despite their opposition to all things Daedric.
The Unofficial Skyrim Patch, version 1.2.6, addresses this issue. They will only carry non-Daedric weapons.
Although the Vigil claim to be against all things daedric, you can conjure Atronachs (such as a frost atronach) in their presence, or have thralls around (like Dead Thrall or a flame one) and they will not be hostile. Same goes for being a vampire or werewolf: they will not recognize you as one, despite their hatred of vampires and werewolves, too.
In the game data, there is dialogue for Vigilants stopping you on the road if you are carrying a Daedric artifact. However, this is set to never happen in-game:
"Stop! We know you're holding a Daedric artifact. Hand it over."
Why should I hand you anything?
"The Vigil of Stendarr cleanses all Daedric filth from Tamriel. Give us any artifacts you have, or we'll purge you like all the others."
You could choose to turn over your artifacts, ending the conversation peacefully:
Fine. Take all the artifacts I have.
"Wise decision. The vaults in the Hall of the Vigilant will guard this against future temptation."
You could also refuse them, at which point they would engage you in combat:
I'm not giving you anything.
"Then prepare yourself to receive Stendarr's Mercy."
The Unofficial Skyrim Patch, version 1.2.6, addresses this issue. They will only carry non-Daedric weapons.