Vatican II revised the LOTH and urged the laity to join priests and religious, who had always prayed it. It is the traditional prayer of the whole Church and well worth looking at, especially these days, when it is so accessible. The Bear and his mate say at least morning and evening prayers (formerly Lauds and Vespers), and often midday and night prayer (Compline). It is part of the rule for Benedictine oblates.
In summary, it is built around a four-week cycle of psalms. Each "hour" will typically have a hymn; three psalms, along with their antiphons; a short scripture reading and response; a canticle; intercessory prayers and the Our Father with collect; and finally, a dismissal.
All may be simply read, or just the hymn sung, or the really ambitious can learn a set of tones and chant it all, like monks in choir. But reading is fine. It takes about fifteen minutes, and you can do it by yourself, with your significant other, or unwary visitors. (We call it a "prayerbush." We always give people a choice, but when it's time, we do let them know what we're about to do and why, then invite them to join. And it cuts down on visitors.)
Universalis is simple. It has all the hours already all set up for you for each day. You want to say Morning Prayer? You pick that and go. The hymns seem to usually be in four-line stanzas, which makes it easy (for us; we're Catholic nerds) to pick one of the eight tones of our archabbey and chant it. Or you can read it, like poetry. You can even order a nice hymnal like Adoremus and substitute your own, more familiar, hymns.
What's nice is you dont have to navigate eight ribbons, learn a lot of rubrics and figure out which week of the psalm cycle you're on. It's all taken care of for you so you can just pray. My mate used her Kindle Fire and I used my Nexus 7 this morning. It's a complete solution.
Divine Office takes a different approach. It has a group of people that actually do the Divine Office every day, so you "pray along" with them while reading the liturgy on your app. There will always be a recording for the hymn, too. It is great for people who must pray alone. It even shows you how many people are praying with you, with their location -- and yours -- appearing as a point of light on a globe.
Some of the readers are a little emotive for my taste, but others will no doubt be moved. A full cathedral choir with organ sounds odd in my cave during the hymn. I would prefer something more intimate. There are a few minor issues like this, but nothing to prevent me from recommending it.
The Divine Office App is even simpler than the Universalis App, but does require an internet connection, unlike Universalis. Between the two, there is no clear-cut winner. If you think you would like to listen to other people praying with you, then you will probably choose the Divine Office app. If you would rather do it on your own and that sounds like a distraction, then Universalis is a better choice.
I would say the Divine Office app is probably better for an absolute beginner, because it gives you a feel for what the prayer is all about. Also, you can just listen. You don't have to know anything. Universalis is good for those with some familiarity with the LOTH. Both are good, which is why both are always a click away here. The Bear's own little campaign to bring the prayer of the Church to others.
The old-fashioned way is to get a copy of Christian Prayer from Catholic Book Publishing Corporation, home of the familiar St. Joseph missal. It is perfect for lay use, and includes a mini-hymnal. It is a nice book with imitation leather cover and zipper. Books just feel better during prayer somehow. The advantage of this edition is that it keeps you in sync with the Church calendar. There's something to be said with knowing the rubrics: what you do and why. There's nothing like doing it the hard way with a book (and maybe a book to teach you how to use your book) to really master the LOTH.
Be sure to get the little white book published each year that tells you how to set up your ribbons! (In the alternative, the Divine Office website has your ribbon placement, too, so that is not as important anymore.) Books are nice, but the Bear cannot tell a lie: they take some study and practice before they cease to be more frustrating than fruitful. Please don't let that bit of honesty deter you. Many worthwhile things take some initial effort. As already hinted, there are even books available to help. The Divine Office for Dodos is charming and instructive.
The Shorter Christian Prayer just has the four-week psalter, and is easier to use, but you miss a lot that you get with the bigger volume. Still, it is excellent for beginners, and has a little bit of seasonal material, although no music. The same company makes the complete four-volume set that includes everything from the office of readings, but that is probably too much for most lay persons (not to mention expensive).
The Litugy of the Hours for Benedictine Oblates is not just for oblates. It is from St. Meinrad Archabbey, and is very simple to use. It has the eight tones used by the monks at St. Meinrad, which you can, with some effort, master with an app like iChant. It helps to have an ear for music. The psalms can be read, but an experienced eye will notice they are chant-friendly: even stanzas with four, five, or six lines. Hymns are included for every prayer. This is the book we use, except for special times, when a four-week psalter just doesn't bring us into the season.
With so many options this a perfect time to learn to pray with the Church. The Divine Office app is 25% off until Ash Wednesday.