While variants of the Cyrillic alphabet have been in use to write Slavic (and some Central Asian, non-Slavic) languages for more than a thousand years, there was another alphabet, so called Glagolitic (from Old Church Slavonic glagol meaning "word"), that was used side-by-side to Cyrillic in the early history of writing in Eastern Europe.
The oldest recorded form of a Slavic language is Old Church Slavonic, which used both Cyrillic (with 44 letters!) and its version of Glagolitic, which looks like this:
Ironically, it seems that it was St. Cyril who invented Glagolitic, not Cyrillic which was named after him. It is thought that students of St. Cyril created the Cyrillic to replace Glagolitic. Maybe a bit of monastic teacher-student competition? :)
Alternatively, Bernard Comrie (of University of Southern California) came up with a more realistic reason for the competition between Glagolitic and Cyrillic. He theorized that Glagolitic came from cursive Greek scripts, while Cyrillic derived from Byzantine Greek uncial scripts already used in manuscripts. The students of St. Cyril might have found Glagolitic "undignified and unsuitable for ecclesiastical use" (Hersey) because of its cursive shapes, and derived Cyrillic from an already liturgical script.
In most places Glagolitic gave way to Cyrillic after the 12th century. In Croatia, though, it continued to be in use until the 19th century in church. The Croatian Glagolitic is quite similar in to Old Church Slavonic Glagolitic, but it has less letters and the shape of its characters are much more rectangular.