Minuscule, or lowercase, refers to the smaller form of letters: a,b,c. Originally alphabets were written entirely in majuscule (capital) letters which was spaced between well-defined upper and lower bounds. When written quickly with a pen, these tended to rounder and simpler forms, like uncials. It is from these that the first minuscule hands developed, the half-uncials and cursive minuscule, which no longer stay bound between a pair of lines.
These in turn formed the foundations for carolingian minuscule, developed by Alcuin for use in the court of Charlemagne, which quickly spread across Europe. Here for the first time it became common to mix both majuscule and minuscule letters in a single text. Traditionally more important letters - those beginning sentences or nouns - were made larger; now they were written in a different script, although there was no fixed capitalization system until the early 18th century (and even then all nouns were capitalized, a system still followed in German but not in English).
Similar developments have taken place in other alphabets. The Greek alphabet was put in minuscule form around the same time that the Roman alphabet was, although uncial letters were not mixed in until about a century later. Ancient texts tend to capitalize mainly proper nouns; the modern practice of capitalizing every sentence seems to be imported.