An alphabet is a writing system where each written character represents a sound. Vowel and consonant sounds are more-or-less treated equal. Example: Greek.
An abjad has each character represent a consonant sound, and vowels are typically not represented orthographically. Examples include the Hebrew and Phoenician scripts.
A syllabary has the entire syllable taken as one unit and assigned a written character. The Japanese writing system is largely an example of this.
A logographic writing system has written characters assigned to morphemes. The relationship between the morpheme’s pronunciation and the assigned written character is hard for me to describe, but with Chinese characters (an example of a logography), graphemes with related pronunciations commonly have similar parts.
I don’t know what an abugida is and thus won’t provide a description, but I do know that Devanagari and the Thai script are examples.
I know I previously wrote a post on the Poneglyph script, but I don’t remember what it said. I’m pretty sure it isn’t an alphabet or abjad, though.