The first category I will cover is higher education. As anyone familiar with LDS education is aware, the Church actually owns and operates several institutions, four of which would fall in the higher education category. These schools are Brigham Young University, Brigham Young University--Hawaii, Brigham Young University--Idaho, and LDS Business College. At this point one might want to mention an institution like the BYU Jerusalem Center for Near Eastern Studies, as it is owned and operated by the Church, and it offers a higher education curriculum. It even has a separate open application process from the other Church schools. The question here is one of administration and autonomy. While non-BYU students may apply to the Jerusalem Center, it is officially under the BYU umbrella in a way that the Hawaii and Idaho campuses are not. Also, if we list the Jerusalem Center separately, we would have to then consider the Barlow Center in Washington DC, BYU's London Center, and maybe even BYU's Salt Lake Center. These programs generally restrict admission to already admitted BYU students, but I've heard (unsubstantiated) stories of students getting admitted to BYU, with their admission being restricted to their participate in the Washington Seminar. To further complicate matters, the Church runs the Missionary Training Center in Provo (in addition to several others around the world), which is officially under the BYU umbrella, and is arguably one of the best language schools in the country, but one cannot apply to attend in the traditional sense, and no certificate or transferable credit is offered, so I'm really going to have to come up with a better definition of what constitutes "higher education" if I hope to come up with a comprehensive list.
Moving on to those programs which operate independent of the Church, but nevertheless hold to an LDS mission, the most prominent and fully operational program is Southern Virginia University (SVU). This school explicitly mentions the Church in its mission statement and it offers a full undergraduate education which has been certified by Virginia and accredited by the American Academy for Liberal Education, an accrediting body recognized by the U.S. Department of Education. The only other fully functional program that explicitly adopts an LDS mission is the Academy for Creating Enterprise (ACE). This group maintains three campuses, one in the Philippines, one in Mexico, and one in Brazil. The basic program is an 8 week course in entrepreneurship. At one time BYU-Hawaii offered a certificate to graduates of the Academy, but their website has changed several times, so it is no longer clear that this is the case.
There is one more functioning program that deserves mention as being in the LDS tradition, but it is not clear how committed it is to the Church of Jesus Christ of Latter-day Saints. This is Nauvoo University. Though Nauvoo University is clearly being headed up by members of the Church, at one time Nauvoo University had a member of Graceland University's administration (a university sponsored by the Community of Christ) on its board, and for a time it presented itself as serving members of all Mormon traditions. This makes sense in that it claims to be a reestablishment of the original Nauvoo University founded by Joseph Smith, but it leaves one to wonder whether it fits in the above definition of an LDS education. They did adopt the original mission statement given by Joseph Smith as their own, but this does not explicitly mention the Church. Recently they appear to have moved closer to a more exclusive approach as to which of the many Mormon traditions they favor, but even so, they are not explicitly LDS in nature as are the Church-owned schools, SVU, and ACE, and one must wonder how their character will evolve in time given how new this institution is as an independent body. Currently they only offer a semester worth of classes, but they are clearly a functioning institution of higher education.
There are three other programs which are explicitly LDS in nature, but are not really functioning programs. The first is Desert Valley Academy (DVA), which has yet to offer any classes, but plans to open in 2012 as a full degree-granting university. I wish them luck, but only time will tell if they are successful. The second program is Bellota a Roble (or Acorn to Oak), which has held GMAT prep courses in Argentina, and currently operates an apartment in Cordoba where LDS students can stay, but they have yet to implement something consistent enough to be considered a fully functioning program. Their stated goal however is to build a full university someday. Finally, a group of BYU students have gone to Japan every summer under the auspices of the BYU Japanese English Education Research Center for the past four years to teach English in some of the Church buildings to members of the Church who need to get their TOEFL scores up so they can attend one of the Church schools in the United States. This group is headed up by Professor Watabe from BYU. He has also been actively working to build support for something more permanent, including possibly an LDS university in Japan so students would be able to enjoy the benefits of an LDS education without having to pass the TOEFL and traveling to America. As of yet, this project has not taken on a formal structure.
Thanks to some wonderful, but somewhat dated, research by E. Vance Randall of Brigham Young University and Chris Wilson of Loyola University published in proceedings of the 12th Annual Conference of the LDS International Society, another group has been identified. These are educational initiatives that were founded by LDS members and in which many or most students are LDS, but which tend to be more general in their mission, often having a general humanitarian purpose. Having worked in Bolivia with Ascend Alliance, I know that most of the students in our business classes were initially LDS, and we actually made a proactive effort to find non-LDS students in keeping with the general humanitarian mission. I haven't verified the operation of all of these groups, but this lists some of them: Choice Humanitarian, UNITUS, JUCONI, Ascend Alliance, Enterprise Mentors International, Reach the Children, Norma I. Love Foundation, Rose Education Foundation, Help–International, American Indian Services, Huntsman Armenian Projects, Ouelessebougou–Utah Alliance, Norman Gardner/Braille Resource and Literacy Center, and Universidad Hispana.
And finally I have heard rumors of yet another school in Guatemala, founded by members of the Church for returned missionaries, but I am still looking up the background information.
This brings me to the end of this list. If I missed anything let me know. Next time I will try to shift to primary and secondary.