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What do you call a language that doesn't use the European alphabet (abcd...), like Mandarin and Japanese?

Is there a word for it, or maybe an adjective that characterizes as being "non-alphabetic"?

I really can't think of a word.

Chinese uses ideograms. It has an ideographic character set. Each character represents an idea.

Ancient Egyptian used hieroglyphs. It had a hieroglyphic character set. Each character was a more-or-less recognizable picture of something.

Most Slavic languages use Cyrillic alphabets. Their characters are letters. Saint Cyril developed the first such alphabet as part of his work converting the Moravians.

Arabic and Hebrew are written in scripts that are named for their languages. They have letters for their consonants. Since medieval times, Hebrew has had diacritic marks for its vowels. Some Arabic vowel sounds are represented by letters; others are optionally represented by diacritic marks.

In technical writing, you should talk about "Logograms" (ie Chinese characters), in which each character represents a word or morpheme. Chinese is a logographic writing system. (The Hieroglyphs of ancient Egypt were partly logographic but mostly phonetic on the "rebus" principle.)

There are syllabaries, in which each character represents a syllable. Japanese kana is an example, and written Japanese is a mixed system with both logograms and two syllabaries.

There are then abugida (such as Devanagari used in India), Abjad (such as Arabic, or Hebrew in which vowels are omitted), and Alphabets in which vowels and consonants are written with separate symbols

Examples of alphabets include the Greek, Latin, Cyrillic and Hangul (Korean) writing systems. Many languages can be written in several different scripts: Turkish, for example, can be written in Arabic or a version of the Latin script.

There's no short way to specify "Languages that don't use Latin script", just as there is no short phrase for "fruit that are not apples".

However, if you are writing about Chinese character systems used for Japanese or Manderine then "Logographic" is the correct word. Technically "Ideographic" refers to systems such as "road signs" in which a symbol represents an idea.