The Monastic Hours are the original formulation of this liturgy for the Roman Rite and follow the guidelines given by St. Benedict in The Rule. These can be found in the Monastic Breviary or The Diurnal.
The 1960 Hours are the Hours which corresponds with the 1962 Mass. Sometimes these are called the Roman Hours. These often come in a three volume set.
The Divine Office is a term which can be used interchangeably for Liturgy of the Hours. However, most often it is used to refer to the 1960 Hours and/or the Monastic Hours exclusively, omitting the newest version. You will find arguments that the terms are different and refer to older and newer forms respectively. Use your best judgement.
The 1970 Hours are the Hours which corresponds to the Mass promulgated in 1970 after the second Vatican council. This version of the Hours and the Mass are referred to as the Ordinary Form. These can be found in full in a four volume set, or partially in the Christian Prayer book and the Mundelein Psalter.
iBreviary is available online, Android, and iOS.
http://www.ibreviary.com/m2/breviario.php
This app has all the offices of the 1970 Hours available in many languages including Latin, Spanish, and English. You can download the prayers and the beginning of the day and access them offline later.
Laudate is an app available on Android or iOS. It has many prayers, complete Catholic Bibles, and more. Overall it is a wonderful app; however, for the Liturgy of the Hours it simply redirects you to the Universalis website. This means you have to have your data or wifi on to access it, and you might as well go directly to the Universalis website.
Universalis is available online, Android, and iOS.
https://universalis.com
This app provides all the offices for the 1970 Liturgy of the Hours.
Free version: The online website uses a non-ICEL translation for the English. This is fine for personal use, or if everyone in your group is using the same app. It can be a problem if some members of your group have a book version or a different app they are using. You will quickly see the meaning is the same but the wording is slightly different making it difficult to pray together.
Paid version: Details on costs can be found on the FAQ page here:
https://universalis.com/n-faq-nothing.htm
You can pay a flat rate that will get you access to the app forever. Or you can choose to pay a monthly rate. The paid version has the same English translation as is found in the 4-book breviary set. The paid app allows you to download the prayers when online and then access them offline.
Magnificat has an app which includes daily prayers for morning, evening, etc. They state the prayers are inspired by the Liturgy of the Hours. These are not the prayers you would find in the 4-book breviary set. This means this would be okay for an individual to use, or if a group is all using the same app, but it will not work in a group setting with people using different sources.
Divine Office app is available online, on Android and iOS.
https://divineoffice.org/welcome
The website and app have the 1970 Liturgy of the Hours. On their homepage they mention having some issues they need to work out concerning copyright permissions. They mostly closed registrations online, but reopened them again to meet the needs of the homebound caused by COVID-19. As of Aug 20, 2020 the app is downloadable for free and does not require registration.
On the app you can play a recording of a group reciting the prayer and / or read the prayer in text.
All Catholic Prayers does not appear to have anything resembling the Liturgy of the Hours (tested on Android in August 2020)