"Shingeki no Kyojin" is most commonly translated as "Attack on Titan". Unfortunately, there is almost nothing alike in meaning between these two titles, and it is a major error, likely made by the author himself or his editors or publicists. The following is an accurate translation of the Japanese title and its significance to the story.
In fact, there are at least three of these in this short title. The most important of these is the erroneous choice to translate "kyojin" as "titan".
When we look into its etymology, the word "titan" does indeed refer to a "being of enormous size", however this is already making an error in one of the first precepts of translation: simplicity. "Titan" may be an excellent word, but it is also a very weighty and complicated one. Originally, the word comes from the name of a Greek mythical race of powerful beings, alike to man but enormous. They attempted to overthrow Zeus but were unsuccessful. One of the more commonly known of these was the Greek hero, Prometheus, who supposedly made man out of clay and also gave us fire. This information, though interesting, is distracting and unrelated to the Japanese word "kyojin".
"Kyojin" is an extremely simple word, consisting of two kanji which translate as "big" and "human" respectively. A more correct translation of this word is simply "giant", because this is our most accurate and simple way to refer to a race of people who are alike to humans but significantly larger. A further error in choosing this word will be described in detail in the grammatical section, but involves the unfortunate fact that "Titan" is also the name of one of Saturn's moons, causing unnecessary confusion, which "giant" would avoid.
One might assume that the author's or translator's intention may have been to create a new term specific to the creatures portrayed in the story, however since this is not reflected in the original title, we find it is simply gilding the lily, making something out of nothing. This is in fact doing a disservice to the original.
The next, and some might say more grave, error is in the selection of the broad and simple word of "attack" as a translation of "shingeki". Though similarities may be found between the two words, the most important emotional and semantic connotations are notably different. "Shingeki" is a very specific term for a military advance, something which is designed to cut through enemy lines using sudden, brute force. The main force of the army follows in the wake of this advance, and that is the author's intention, that is the image he seeks to draw. "Attack" is not only overly simple, but it leads to an error in interpretation, especially when combined with the preposition "on" and the noun "titan", which again, shares its name with something else in common knowledge.
Finally, the word "on" is not in the original title at all and is completely unnecessary. The reason for its presence is explained in the grammatical section.
So far, what has been explained is that at least two words in this title were mistranslated, but the word order and presence of the preposition "on" are still misleading. If left in this fashion, the new title becomes "Advance Giant", which is obviously incorrect, not to mention nonsensical.
As discussed earlier, with the current, erroneous title, there is a very unfortunate error in interpretation which occurs, at the hands of our "on" and the fact that "Titan" is a moon orbiting Saturn. In fact, the most natural interpretation of this title is perhaps, "an alien attack on one of Saturn's moons", though of course anyone who had seen screen shots or cover images of the series would presume this to be incorrect. The next most natural assumption would be that "Titan" is a place, but somewhere on land, somewhere capable of being attacked. Finally, that it is either a person or an organization, but this is less natural. Unfortunately, none of these are even close to the true meaning of the title.
We can gather from observation that "titan" does not refer to a place, person or organization, but to the race of giants which threaten the heroes. Any native speaker of English will attest that the grammar of "Attack on Titan" makes no sense, given this knowledge. Of course, the word order is partly to blame, but the main problems come from the simple preposition, "on".
An attack "on" something means that whatever "on" refers to is the thing being attacked, not doing the attacking. "On" is an extremely passive preposition, implying in fact stomping, setting things upon or, even at its most basic, simple oppression. As we are aware from observing the series, none of these interpretations are accurate.
The reason for its presence here has to do with a Japanese function word known as a "particle", which is used to connect nouns to verbs, to define them and differentiate between different types of verb referents. We have no similar class of word in English, but we often use prepositions in a similar way, thus Japanese people who learn English as a second language tend to overuse prepositions in places where particles would normally go. Unfortunately, as you can imagine from the sound of it, particles are extremely prevalent in the Japanese language. Even in sentences which do not possess particles, they are almost always implied. English on the other hand can get by quite often without prepositions. As an example, a sentence like this: "I put the can there", if translated into Japanese, would result in at least two and potentially three particles (e.g. watashi WA kan WO asoko NI oita), while English needs no preposition in this case. Thus, though often crucial, prepositions are much less common in English than particles are in Japanese.
The Japanese word "no" (as in Shingeki no Kyojin) is something that is often translated as either the preposition "of" or the possessive "(apostrophe) 's". It is a marker of possession. In this case, "kyojin" is something which belongs to "shingeki". From the previous semantic changes, we can say that the giant in this case is something that belongs to the advance, a giant which is of or pertaining to a military advance. Now we are finally approaching the correct title.
In fact, what the author intended was something quite beautiful, which is completely lost in this incorrect translation. The correct translation might go as follows, "Shingeki no Kyojin" / "Giant of the Vanguard". [spoiler] The title in fact refers to Eren Jeager himself. 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. This title is considered, sensitive and contains a high emotional value, especially when one is aware of all the circumstances involved in the story. "Attack on Titan" is not only completely incorrect, but doing a great disservice to both the author and the story.In conclusion, once again, what the title actually refers to is not an attack on anything, but rather the giant who is finally able to lead the human race out of darkness and toward victory.