The Office of Readings
The Office of Readings was created by the 1970 reforms of the Liturgy of the Hours.
Originally the office of Vigils was created to be recited in the middle of the night - not just before going to bed, as with Compline, but by getting up in the dark to recite it.
This proceeding being found to be too heroic except for some monasteries, Vigils began to be said first thing in the morning, just before what was then called Matins. In due course the name "Matins" became applied to the whole combination, and the second half (which had been called Matins to start with) began to be called Lauds.
Matins, the Catholic Encyclopaedia says, was "the most important office of the day and for the variety and richness of its elements the most remarkable". After an invitatory psalm that served as an introduction to the whole Divine Office of the day, there followed twelve psalms and three readings on ordinary days, and eighteen psalms and nine readings on Sundays. This was indeed rich and varied, but it was also too much for human frailty and certainly impracticable for the laity to find time for in the mornings, especially if it was immediately followed by Lauds.
The 1970 reforms prune Matins to a more digestible three psalms and two readings. More importantly, Matins is also purged of any relation to the time of day, and it may now be said at any time at all, whatever is most convenient. It is even possible to say, in the evening, the Office of Readings for the following day.
Of course there is no reason not to continue having the Office of Readings in its old place as the first Hour of the day, but the reform makes it easier for people to fit it wherever there is room for it in their daily routine.
The Invitatory Psalm has accordingly been detached from the Office of Readings, and is directed to be said before whichever Hour you recite first on a particular day.
The structure of the Hour
The Office of Readings starts with three psalms (or parts of psalms) with their antiphons.
Then comes a reading from Scripture: the readings may repeat every year or every two years. The printed books use a one-year cycle, and so does the Universalis web site. The Universalis apps and programs can use either the one-year or the two-year cycle.
Then comes a second reading, which may be from the Fathers of the Church or may be by or about the saint of the day.
On Sundays and high feasts, the Te Deum follows.
Finally, a concluding prayer brings the Office of Readings to an end.