The direct English translation of 進撃の巨人 ("Shingeki no Kyojin") isn't actually "Attack on Titan", and the name seems like an odd translation because it implies either an attack on a person/thing named "Titan" or an attack on or taking place on a planet/place called "Titan," neither of which are the case here. Have there been any official announcements on the translation/English naming of "Attack on Titan" regarding why they gave it that name?
To answer your questions, one must note one important thing:
- The original English "subtitle" for 進撃の巨人 ("Shingeki no Kyojin") is in fact "Attack on Titan." (See the volume one cover for reference.)
Semantically there is very little meaning that would connect the two titles, and is likely to be attributed to an error, on the part of the author or his editor/publicists.
The biggest notable change is the use of "titan" as a translation for "kyojin."
According to the Merriam-Webster dictionary, titan is defined as:
capitalized: any of a family of giants in Greek mythology born of Uranus and Gaea and ruling the earth until overthrown by the Olympian gods
one that is gigantic in size or power: one that stands out for greatness of achievement
In terms of simplicity, titan might be an very good choice of words, but in terms of conveying the actual meaning, it falls a bit short.
"Kyojin" is a very simple word, consisting of two kanji which translate as "gigantic" and "human" or "person" respectively. Naturally a more correct translation of this word would be "giant", because it's simplest way in English that we can to refer people are like humans, but are significantly bigger.
"Attack" seems like a broad and simplistic choice of "attack" for "shingeki." both share similarities to one another, the semantic connotations are a bit different. "Shingeki" is a very specific term for akin to a (swift) military advance/charge on the enemy. Using "attack" oversimplifies this, and in combination with the preposition "on" and the noun "titan" (which can also refer to one of Saturn's moon by the same name), leads to an error in interpretation. Shingeki can also be used to refer to a vanguard rally/charge, much like what the Survey Corps are.
But let's take a deeper look at the title itself and see what we can get from it. It we directly translate and take it's meaning literally, the title would be something like "Advancing Giant." This doesn't work well as a title and probably isn't what I think the author intended on convey to his readers.
If we read the English titles as it is, "Attack on Titan," grammatically one would infer that "Titan" is some sort of location rather than a person or group. E.g., "Attack on Normandy." For this we have the preposition "on" to blame. Initiating an attack "on" usually implies that whatever "on" refers to is the thing being attacked, not doing the attacking. So the combination of "attack" and "on" as a passive proposition, implies in the act of inciting an act of aggression upon the following noun, "titan."
The Japanese particle "no" (in "Shingeki no Kyojin") is often seen translated as either the preposition "of" or the possessive (-'s), as a marker of possession.
In the case of the title, "kyojin" would be something possessed by "shingeki". Therefore we can assume that the giant is something that belongs to the advance/charge, in other words a giant pertains to a military advance/charge of a vanguard.
Now what does all this mean? Is it referring to the Titans as a whole or singling out on in particular? One can imply that the title is referring to Eren Jaeger, as he is the one and only giant who fights on the side of the humans, thus the giant who leads the way for their armies to finally make a dent in the enemy forces, representing hope for the humans. So something like "The Advancing/Charging Giant," or more appropriately something "The Giant of the Vanguard." Essentially referring to the giant that would lead people towards victory/salvation.
The original title was most likely intended to convey a great deal emotional value, especially when the circumstances involved in the story. However, this might not be as easily conveyed in English so they often to go with something short and to the point for the English subtitle (which is typically only there for the "cool" factor) and never intended to be used as the localized name.
For better SEO, sometimes localizers like to include the romanized Japanese name or w/e the "engrish" subtitle might be, like in the case of Oreimo, this way they can bring already associating links from the original source material to their localized version, without needing to break the bank with advertising.
With the lack of a good answer i'll flesh my comment out.
Google Translate translates 進撃の巨人 ("Shingeki no Kyojin") to Giants Advance. now Google Translate is machine translation so not good with sentances but it does does help with single words as guidelines, so the 2 main parts of the titles are
Now の (no) gets translated to of so putting all together we get Advance of Giant.
Now why did the title then become Attack on Titan, well without any explanation of the translators we can assume the words was changed to make the title sound cooler as per Eric's (Listed as キルア) said in the comments to the question
it's just a typical grammatical abomination as a result of someone trying to make a cool and catchy English title
so how did they do that? well we can assume that Advance and Giant were changed to Attack and Titan but why changed Of to On, the only reason i can think of is in the original title Shingeki no Kyojin, if you did a partial translation it could be Advance no Giant, someone proably went and reversed no to on so we get Advance on Giant, do the same word replacement as mentioned above and we get Attack on Titan
Now this answer relies heavy on Google Translate, however this answer also applies logic, you can find better explanation of Japanese Language here, while it properly translates the original title and explains the use of の in it, it does to explain how Attack on Titan came about so this answer serves to add assumed logic behind such a decision.