The ess-tsett ("ß") is a letter used only in the German alphabet. It represents the ligature "ss" under certain conditions (cf. the letter W, which represents a ligature, too: "double u"). "ß" is unique among the letters of western alphabets in that it has no majuscule; "SS" must be used in an all-caps environment. "ß" should not be confused with the lowercase Greek letter beta ("β"), which it resembles but to which it is unrelated.
The name ess-tsett is a phonetic circumscription of how the two letters "s" and "z" are pronounced in German. Although the letter is universally called "sz", this is historically imprecise, since it originally derived from a Fraktur character representing the ligature of the long or medial "s ("ſ") with the short or terminal "s" (now the conventional minuscule letterform) -- that is, "sz" is really "ſs".
In today's German orthography, "ß" (like other "simple" consonants) is used after a long vowel, while "ss" (like other "doubled" consonants) is used after a short one. Both represent the sound /s/; a solitary "s" has the value /z/ (although this is devoiced at the end of a word). For example, Fuß (/fu:s/, German for "foot") has a long vowel, while Fluss (/flUs/, meaning "river") has a short vowel (cf. the difference of engl. "c(e)" and "ss" in "mice" and "miss").
Until the reform of German orthography (Neue Rechtschreibung) in 1998, an additional rule prescribed that "ss" would never be used at the end of a word and be replaced by "ß", even if it followed a short vowel. As a result, Fluss was formerly spelled Fluß; the new rule gets rid of the irregularity that even under the old orthography, the plural Flüsse was spelled with "ss" because in the plural, the "ss" was not at the end of the word. This is to accord with the orthography of other consonants, which are single after a long vowel and double after a short one; for example, egal /ega:l/ with a long "a" and Ball /bal/ with a short one.