The Cyrillic alphabet was traditionally one of the two scripts invented to write Slavic languages, the other being Glagolitic. However, Cyrillic emerged as the more widespread of the two, probably due to its similarity to the Greek alphabet. At its height during the Soviet Union, Cyrillic was used to write not only Slavic languages such as Russian, Byelorussian, Ukrainian, Serbian, Macedonian, Bulgarian, etc., but also languages from other family like Mongolian, Uzbek, Kazakh, Azeri, Tajik, and so on. After the breakup of the Soviet Union, many of these languages have started to move toward other alphabets, such as Arabic and Roman.
It is rather clear to see that a majority of the Cyrillic letters were derived from Greek. The backward N which stands for /i/ comes from Greek eta η. Some of the letters (such as sibilants) were borrowed from Hebrew and Syriac alphabets.
Quick note: Ъ and Ь used to represent a very short high back vowel and a very short high front vowel, respectively. They are now silent but mark the quality of the consonant preceding it. Ь means that the previous consonant is palatalized, while Ъ means that the preceding consonant is not palatalized even in a palatalizing environment. (Palatalization is the tendency of vowels like [i] and [e] to push the tongue toward the front of the mouth while pronouncing a preceding consonant, causing the consonant to change a little, like [t] to become [ts], [s] to [š], etc).
The following is the modern Cyrillic alphabet as adopted to write Russian. Each of the Cyrillic letters is actually a pair, the upper-case letter on the left and the lower-case letter on the right. Roman letters in blue represent traditional transliteration of Cyrillic. The purple letters are the phonetic value of the Cyrillic letter.