MOGAI stands for Marginalized Orientations, Gender Alignments and Intersex. The acronym was coined as an alternative to LGBTQIAPN+, for being shorter and more inclusive.
Marginalized Orientations are orientations that aren't the default. For instance, someone whose sexual orientation is not heterosexual, or someone whose romantic orientation is not heteroromantic.
Gender Alignment, while usually meaning something else nowadays, was used to mean that someone's birth assignment didn't align with the person's actual gender. 
Intersex means someone who doesn't fit the norm of what society considers a "male body" or a "female body", without requiring any sort of treatment or surgery.
Not much is known about how exactly things went, since a lot of blogs involved are now deleted, and the whole thing happened on different reblog chains. However, what we know is that MOGA on its own was coined first, and later on, Intersex was added. Cloud, Tumblr user cisphobeofficial, is credited with adding the final I.
Since the addition of Intersex, some people mistakenly spread that the A on the acronym stands for And.
Another development is that some people have ended up using MOGII, with GI standing for Gender Identities. A lot of people are against this, though, because that would be saying certain women would qualify as MOGII when they aren't of a marginalized orientation, transgender or intersex, while transgender men who aren't intersex or of a marginalized orientation could be left out because they identify as a privileged gender identity.
Some people also use IMOGA, which holds basically the same meanings, but rearranged so Intersex stands out more.
The acronym COGAP is based on MOGAI, to be an alternative to dycishet or pericishet. COGAP stands for Centered Orientations, Gender Alignments and Perisex. 
Several groups try to paint MOGAI as a "bad acronym". 
One of the reasons is supposedly that MOGAI covers more than it should, including philias and kinks. Those claims have no proof attached to them, and are usually dismissed.
Because it doesn't center any identity in particular, people against the acronym use it as a substitute for identities they find "too specific" or "ridiculous", saying only certain identities are MOGAI.    
- ↑ https://autismserenity.tumblr.com/post/143866147761/what-is-mogai-ive-always-heard-about-it-but-i
- ↑ http://bluandorange.tumblr.com/post/72118874359/infernalteuthis-davekat
- ↑ http://miscellanii.tumblr.com/post/77534040363/ive-seen-another-short-hand-for-lgbtquia-was
- ↑ http://cisphobeofficial.tumblr.com/post/120654835099/hey-cloud-youre-the-one-who-created-mogai-right
- ↑ http://cisphobeofficial.tumblr.com/post/85767512954/wait-wait-mogai-doesnt-mean-marginalized
- ↑ https://fuckyeahasexual.tumblr.com/post/120668684488/this-explains-the-issues-with-gsmgsrm-and-the
- ↑ http://cisphobeofficial.tumblr.com/post/86223689439/what-do-you-think-of-the-grsm-acronym-gender
- ↑ http://cisphobeofficial.tumblr.com/post/89066294324/i-apologize-if-youve-explained-this-before-but
- ↑ https://autismserenity.tumblr.com/post/153172851931/what-does-mogai-mean
- ↑ https://autismserenity.tumblr.com/post/143865184426/definitions
- ↑ http://gaypeachs.tumblr.com/post/168877698016/why-do-people-hate-the-term-mogai-so-much-from
- ↑ http://cisphobeofficial.tumblr.com/post/120825816794/apparently-the-creator-of-mogai-is-a-pedophile-who
- ↑ https://autismserenity.tumblr.com/post/160626232011/perrywingr-something-that-ive-been-sitting-on
- ↑ http://bi-privilege.tumblr.com/post/102405678095/apparently-using-mogai-instead-of-lgbt-is
- ↑ http://aphobe-nonsense.tumblr.com/post/152600052901/discourserose-aphobe-nonsense
- ↑ http://apersnicketylemon.tumblr.com/post/166659558797/hey-do-u-have-any-sources-about-where-the-mogai
- ↑ http://gaypeachs.tumblr.com/post/168873001051/aphobicbxtch-gaypeachs-mogai-is-fucked-up
- ↑ https://a-polite-melody.tumblr.com/post/167710772137/random-question-but-what-nb-identities-are-mogai