The Cyrillic alphabet was created in the 9th century CE, having been commissioned by the Bulgarian tsar to supersede the earlier Glagolitic script. It was based largely on the Greek alphabet, with some letters from Glagolitic used to represent sounds that existed in Old Church Slavonic but not in Greek. The script is named after Saint Cyril, who was one of the two brothers who created the earlier Glagolitic script.
The script is particularly widely used across the eastern and southeastern Slavic languages, where it is thought to be optimised for those languages’ phonologies (with letters like ь and ъ representing “soft sign” (palatalisation of a preceding consonant) and “hard sign” (inserting a break to prevent palatalisation of the preceding consonant) respectively). It’s also used widely for non-Slavic languages that fell within the USSR’s sphere of influence, and historically in many languages that fell into the Orthodox Christian sphere, like Romanian (which is not written in Cyrillic now, although in Moldova Cyrillic was officially used until 1989). Some languages that are currently written in Cyrillic, like Kazakh and Mongolian, aspire to transition to alternative alphabets (Latin and traditional Mongolian, respectively) by 2025.
The Cyrillic script seems to vary more across the languages that use it than, say, the Latin alphabet does. Comparing, for example, Russian and Serbian alphabets, each one includes letters not used by the other. Their alphabets also include different numbers of letters (33 in Russian vs 30 in Serbian). As a semantic note, Wikipedia uses the term “Cyrillic script” for the letter set as a whole, while “alphabet” seems to be preferred for a specific language’s implementation of Cyrillic.
There are a number of systems in use for transliterating between Cyrillic and Latin alphabets. For example, for Russian alone there are at least 13 separate systems, as Wikipedia can tell you . In Serbian, where officially both alphabets are equally acceptable, there is a 1:1 correspondence between Cyrillic and Latin graphemes. The constructed language Interslavic also has a systematic correspondence between Cyrillic and Latin representations of the language, although it has some Cyrillic letters represented by Latin digraphs.