If the vowels are optional or not present, e.g. there's one "k" symbol regardless of the vowel, it's an Abjad. The archetypal Abjad is the Hebrew writing system.
If the vowels are written by adding them to the consonant symbol (similar to diacritics), it's called an Abugida. One example of this is the Ge'ez script in Ethiopia.
Generally you look at what concepts are embodied in the script, and at the form of the glyphs. So:
You might have a script that assigns glyphs to phonemes. ("Language is made of sounds.")
You might have a script that assigns glyphs to consonants and doesn't bother to represent vowels. ("Language is made of sounds, and some of them are more important than others.")
You might have a script that assigns glyphs to syllables. ("Language is made of things you can say.")
You might have a script in which the glyphs assigned to syllables are composed of recognizable and conceptually distinct parts, but those parts have no independent representation. (Compare the glyphs ሀ ለ ሐ with the related glyphs ሄ ሌ ሔ.) ("Language is made of things you can say, but there are patterns.")
You might have a script that assigns glyphs to words, though in almost all cases you don't. The label "logographic script", applied to a script the labeler doesn't know well, is infinitely more popular than the concept "logographic script". I don't think any script has ever existed meeting the criterion of "it does not represent phonetic, but semantic units". But there are some, and used to be more, that leaned more or less strongly in that direction.