In the Japanese writing system, there are two kinds of characters - the ideographic characters, called kanji, which represent an idea, or unit of meaning, and the syllabic characters, called kana, which each represent a single syllable, but are usually not associated with any specific meaning.
In Modern Japanese, there are two main systems of kana in use today, either of which can be used to represent the all of the currently extant syllables - hiragana (平仮名), which is used to write verb and adjectival inflections and gramatical particles, and katakana (片仮名), which is used to write foreign loanwords and onomatopoeia. Hiragana is frequently thought of as a "cursive" syllabary, because of its rounder, more flowing appearance, while katakana is recognizable by its hard, clean strokes and sharp angles.
Hiragana and katakana are not the only kana systems, however. The first systematic attempt to use Chinese-style characters to represent native Japanese syllables appeared in the Manyoshu ("Collection of Ten Thousand Leaves"), a collection of poetry compiled around AD 759, and is now known as manyogana (万葉仮名), after the title of the work. This crude system of matching Chinese character readings to native Japanese pronunciation slowly evolved into a series of more simplified writing systems, and these intermediary kana systems are now known as sogana (草仮名). Sogana would ultimately evolve into the two standardized syllabaries we now know as katakana and hiragana, but historical texts contain nonstandard kana, or kana from earlier systems that have since died out, often refered to as hentaigana (変体仮名), which can be loosely translated as "weird kana."
Finally there are speciallized names for different ways to use modern kana - small kana written in the margin to gloss the pronunciation of difficult kanji are called furigana (ふり仮名), while the kana that mark case and aspect inflections in Modern Japanese writing is frequently referred to as okurigana (送り仮名).