First I should make it clear that we're not 100% positive that this is the name of Goku's new form. It's definitely the name of the technique of moving without thinking, which was foreshadowed in ROF, but are the form and the technique synonymous? It doesn't seem to make a lot of sense; it's doubtful that the gods and angels who master this technique transform when they do so.
That said, Toshio Yosh*taka (one of Super's writers) promised us that the name would be clever, and in his recent tweets hyping the special he seems to treat it as the name for the form.
The official Toei translation is "Ultra Instinct", but as Herms makes clear, that translation is missing a lot of nuance. We have been rather spoiled up to this point because forms have always had English names. They're not that interesting, but at least they're perfectly understandable to us without any translation at all. This one is different, and that's something we'll have to get used to.
The form’s name in Japanese:
身勝手の極意
Migatte no Goku’i身勝手/migatte typically means “selfish”, although when you break it down, 身 means body while 勝手 can mean “of one’s own accord” (as in 勝手に). With all the emphasis on Goku’s dodging and fast reactions in this state, and the way this seems to tie in with Whis’s teachings about getting each part of one’s body to move on its own, it seems they had wordplay in mind and that in context 身勝手/migatte is supposed to suggest one’s body moving on its own. With that in mind, it seems this is the part of the name which the official subtitles translate as “instinct”.
極意/goku’i means the deepest level of an art or skill, secret teaching, innermost secrets…etc, etc. Basically a fancy way of saying a secret, something known only to the masters of an art and not to the newbs. That’s its typical meaning as a set phrase, anyway. When you break it down though, 極/goku means “extreme” and 意/i means “thoughts” or “mind”. In that light, there may be more wordplay intended here, with the term in this context meant to suggest “extreme mind”, which also tie in with Goku’s increased reaction speed, if he’s in a heightened state of awareness. Anyway, with 極/goku means “extreme”, it seems this is the part of the name which the official subtitles translate as “ultra”.
Keep in mind that this 極/goku is unrelated to the name of our hero 悟空/Gokuu (written with two characters roughly meaning “Aware of the Void”, and with kuu here having an elongated vowel sound). With all the other potential wordplay going on in the name, it’s hard to completely rule out an intended connection, but take it from me that in Japanese 極/goku and 悟空/Gokuu are quite different.
The の/no in the middle there is a possessive or linking particle; precisely how you translate it is highly context dependent. If we translate 身勝手/migatte as “selfish” and 極意/goku’i as “secret”, then の/no could be translated as “of” or as an apostrophe: so you get “Secret of Selfishness” or “Selfishness’ Secret”. The official subtitles leave it out entirely, presumably taking the relationship between the two parts of the name as implicit, and so we get “Ultra Instinct”.
Dokkan Battle and some other merchandise have also referred to Goku in this state as 身勝手の極意“兆”. Now, 兆/chou means “trillion”, but 兆し/kizashi means an “omen”, a sign of things to come. Since Goku has not yet fully mastered this state but presumably will in the future, it seems that they are implying this temporary transformation seen in ep.110 is an omen of the fully mastered state to come, in which case translating this term as Ultra Instinct “Omen” seems appropriate. Unless they were going for “trillion”, though it’s less clear what that would mean in this context. I guess we’ll have to see how this is handled in the English version of Dokkan Battle.
@Herms98
So, how exactly did Goku transform? Here's Whis's explanation from Episode 110:
16:10 | Kai: I can't believe someone who could gain on Jiren this much even exists!
16:15 | Belmod: It's not possible! It can't be possible!
16:19 | Zenō: Goku suddenly got stronger!
16:25 | Future Zenō: Goku suddenly got amazing!
16:28 | Zenō: Why? Why?!
16:29 | Future Zenō: How come? How come?!
16:32 | Daishinkan: Perhaps...
16:34 | Whis: Perhaps something quite wonderful is happening to Goku-san.
16:42 | Tenshinhan: Do you know what it is?
16:43 | Whis: Yes! Though this is just my assumption, of course!
17:00 | Beerus: Ultra Instinct... [gods gasp and gape]
17:11 | Anato: I see. Those movements... But, for a mortal like Son Goku...? That is a state not even a god can easily attain.
17:23 | Beerus: Ridiculous! Impossible, impossible!
17:27 | Roshi: I'm not sure why, but the other gods appear to be getting worked up.
17:40 | Future Zenō: Everyone seems surprised.
17:43 | Zenō: Apparently it's super amazing.
17:47 | Daishinkan: Well, it may be a little early to draw a conclusion. Regardless, this development is captivating.
17:56 | Future Zenō: How exciting!
17:58 | Zenō: How thrilling!
18:00 | Whis: My, my. Who would have thought this to be possible, Beerus-sama?
18:06 | Shin: It seems to be something rather well-known amongst the gods.
18:11 | Elder Kaiōshin: Wh-What, you don't know about it? You amateur.
18:16 | Beerus: But why did it happen so suddenly?
18:19 | Whis: I believe the Genki Dama was the trigger.
18:23 | Krillin: You mean he powered up by absorbing the Genki Dama?
18:28 | Tenshinhan: But the power of the Genki Dama didn't work on Jiren. Even if that power boosted him, it doesn't explain how he can match him.
18:37 | Whis: Precisely. The energy of the Genki Dama must be acting as a temporary source of power for his depleted body. However, the main point is something else. In that big explosion, the raging power of the Genki Dama that entered Goku-san and Goku-san's own power to resist it... Beyond that clash of power, Goku-san broke through the shell to the deeper potential within himself. That's my assumption.
19:13 | Krillin: Broke through the shell...
19:16 | Whis: That's all fine. However... That heat... I'm not sure about this. [Cue this scene.]
20:29 | Jiren: Saiyans... Quite interesting beings. But... [Goku's form fades] This heat... That is your limit. [Jiren knocks Goku out]
What is this "heat"? We have no idea yet, but it will probably be explained in the coming episodes. It's almost certain that Goku will regain this form eventually. It was teased in the final narration:
21:57 | Narrator: The astonishing power Goku displayed. Can Goku again reach that state that even gods fear? The time until fate is decided in the Tournament of Power is 25 minutes.
(Of course he can.)
