张 is a classifier usually used with objects like: 1. which can be sctreched, espacially bow or crossbow, 一张弓 2. which can be spread out, 一张纸，一张桌布 3. which has a flat surface, 一张床，一张椅子，一张木板 (when you say一张木板 it suggests the board is large, or you say 一块木板) 4. furs or skin which has been peeled, this use has been extended to "face" and some other related objects, 一张皮，一张脸，一张面具 The very original meaning of 张 is to stretch. Hope that can help you.

We use 张 to describe the quantity. You can take it as unit, EA, piece. Same as in 一张床

I think it's used to describe something got an area, like the " 桌子" has a surface,

Its use is similar to 'cup of/杯' in 'a/one cup of tea/一杯茶', 'head of/头' in 'a head of cattle/一头牛', 'pair of'/双 in 'one pair of shoes/一双鞋子' and 'stem of/枝' in 'a stem of rose/一枝玫瑰'etc. Sometimes, it's called classifier/分类词, a word or morpheme, as in Chinese or Japanese, that indicates a class of nouns and regularly accompanies any noun of that class in certain syntactic constructions, as in numeration.

it's a measure word for tables 桌子