It's called the Office of Vigils. This is not a daily liturgical hour, but rather a once-a-week Sunday office that is prayed in combination with the Office of Readings on Saturday night. As a key to understanding what Vigils is all about, think about Easter Vigil mass. There's lots of extra readings, right? The office of Vigils is a set of extra biblical antiphons, canticles, and a gospel reading, which, combined with the Office of Readings, prepares us for the "little Easter" that we experiecne every Sunday of the year.
If you have a four-volume breviary, Vigils appears near the end of the breviary in Appendix One.
The idea is to first do the Office of Readings up to the end of the second reading. Then do Vigils-several canticles with antiphons and a gospel reading--and then conclude with the Te Deum and concluding prayer.
If you use ibreviary.com on your mobile device or computer, then Vigils will automatically follow the Office of Readings on Saturday night. I believe this solves my puzzlement over why ibreviary.com switches to the next day's prayers after 6pm each day. This is necessary if they want to give us the Office of readings, and Vigils, as, well, the vigil of the following calendar day.
Since it's been my custom to say the Office of Readings before Morning Prayer each day, this Vigils business hasn't worked out for me. But maybe I'll try to work it in for the next few Sundays of lent and see how that goes.
One more bit of trivia: primitive observance orders, such as the Carthusians, pray vigils late at night. If you have the dvd of Into the Great Silence, you can enjoy watching and listening to the monks praying Vigils on the extra features disk. (In Latin chant, with English subtitles.)