A weak syllable is also called "an unstressed" syllable, and it's the part of the word (or sentence) which is less pronounced when we speak.
So in the sentence "That's a great idea", the word "a" and the syllable "a" on the end of "idea" are weak. However, the words "that's" and "great", and the syllables "i" and "de" from "idea" are strong, and you can hear them more easily.
Very often, weak syllables contain the schwa sound (sounds like "uh") so that we can say them more quickly in sentences.