The word über comes from the German language. It is a cognate of both Latin super and Greek ύπερ (hyper), as well as English over (as in "overkill"). During the late 1990s, über also became a synonym for super; e.g. übercool = supercool, generally with a slightly intensified meaning. Über is commonly written as uber in English, though with slightly different meaning.
The normal transliteration for the 'ü'-Umlaut when changing from German would be ue, not just 'u'; however, it could be argued that the American English use of the word uber is a new word distinct from ueber. This is because English is defined by common use of words, which dictionaries and academia record, not the reverse. The use of 'ü', 'u', and 'ue' in the word is an emerging trend in common usage with no clear consensus.
Uses in German
In German, über is used as a prefix as well as a separate word. In the latter case, it may be a preposition or an adverb depending on context. Eg. über etwas sprechen - speak about something, über die Brücke - over the bridge, übernehmen - take over (nehmen = take).
Über also translates to over, above and super. The actual translation depends on context. One example would be the term Übermensch from Friedrich Nietzsche, which translates to super human or superman (literally "overhuman"). It is assumed that it is via this translation that the word über entered the English vocabulary (cf. calque).
In German online, slang, it can also be used with the same meaning as the English usages of "uber" by gamers. In this case, it is typically written without the umlauts—despite the ready availability and familiarity of the 'ü' character—in part to distinguish it from its original meaning. Linguistically speaking, it's interesting to note that a language can borrow a loan word that it already loaned out, as long as the meaning has changed sufficiently. In the most cases of this usage, it is borrowed directly as Denglisch, or an English word/phrase that has been fitted adhoc into the German language.
