The German alphabet consists of the same 26 letters as the modern Roman alphabet (see more about roman alphabet here). The German language additionally uses three letters with diacritics and one ligature:
ä, ö, ü / Ä, Ö, Ü
ß (called eszett (sz) or scharfes S (sharp s) )
Although the diacritic letters represent distinct sounds in German phonology, they are almost universally not considered part of the alphabet. Almost all German speakers consider the alphabet to have the 26 cardinal letters above and will name only those when asked to say the alphabet.
The diacritic letters ä, ö and ü are used to indicate umlauts. They originated as a, o, u with a superscripted e, which in German Kurrent writing was written as two vertical dashes. These two dashes have degenerated to dots, and look like, but are not a trema.
When it is not possible to use the umlauts, for example, when using a restricted character set, the umlauts Ä, Ö, Ü, ä, ö and ü should be transcribed as Ae, Oe, Ue, ae, oe and ue, respectively; simply using the base vowel (e.g. u instead of ü) would be considered erroneous by German speakers and is prone to producing ambiguities.
Nevertheless, any such transcription should be avoided when possible, especially with names. The reason for this is that names often exist in a variant which uses this style, e.g. "Müller" and "Mueller". In a text which uses this transcription system, it would be obvious that if a person's occupation is given as "Mueller" (a miller), that should actually be spelt "Müller", but for a person whose name is given as "Mueller", there would be no way to tell if the name needs to be back-transcribed or not.
Automatic back-transcribing is not only harmful for names. Consider for example "das neue Buch" (the new book). This should never be changed into "das neü Buch". Technically, the second e has no connection with the u at all: neue is neu (the root for new) followed by an e. The word neü does not exist in German.
Furthermore, in northern Germany, there are family names and place names where e lengthens the preceding vovel, as in "Laermann", which is pronounced with a long a, not an ä.
In proper names there rarely may also appear an ë, which is not an umlaut, but a trema to distinguish what could be a digraph as in French, like oe in Bernhard Hoëcker (although in this case the trema was added by that person).
Swiss typewriters and computer keyboards do not allow easy input of uppercase umlauts (nor ß) for their positions are taken by the most frequent French diacritics. The decision to drop the uppercase umlauts is due to the fact that uppercase umlauts are less common than lowercase ones (especially in Switzerland). Geographical names in particular are often written with A, O, U plus e — despite "Österreich" (Austria). This can cause some inconvenience since the first letter of every noun is capitalized in German.
Unlike other languages (e. g. Hungarian), the actual form of the umlaut diacritics, especially when handwritten, is not all that important, because they are the only ones of the language (including the dot on i and j). They might look like dots ( ¨ ), acute accents ( ̋ ), vertical bars ( ̎), one horizontal bar/macron ( ¯ ), a brevis ( ˘ ) (which was also used to distinguish a "u" from an "n" in some Kurrent-derived handwritings), a tiny N or a tilde ( ˜ ) etc.
Also, the eszett or scharfes S (ß) is used. It exists only in a lowercase version since it can never occur at the beginning of a word (there are a few loan words starting with an s followed by a z (e.g. Szegediner Krautfleisch but that is not the same as the eszett which counts as one letter).
In all caps it is converted to SS, while in Switzerland ß is not used at all, but ss instead. This gives rise to ambiguities, albeit extremely rarely; the most commonly cited such case is that of "in Maßen" (in moderation) vs. in Massen (en masse). For all caps usage, an uppercase ß had been postulated since 1879 and was officially introduced in 2008 into Unicode 5.1 as U+1E9E, although a definite form hasn't been found yet.
Regulations introduced as part of the German spelling reform of 1996 reduced usage of this letter for Germany and Austria (see ß). Although nowadays substituted correctly only by ss, the letter actually originates from two distinct ligatures (depending on word and spelling rules): long s with round s ("ſs") and long s with (round) z ("ſz"/"ſʒ"). Some people therefore prefer to substitute "ß" by "sz". By official rules this is incorrect.