Tavolo, tavola, tavolino
What are the meanings of these words, they can't all be table?so would a chopping board be called una tavola?
what would a "dining table" be called una tavola or is there a grander word?
in English "a kitchen table" is not just the table in the kitchen but suggests the thing around which family resolve issues, is there a similar term?

Question

Tavolo, tavola, tavolino
What are the meanings of these words, they can't all be table?so would a chopping board be called  una tavola?
what would a "dining table" be called una tavola or is there a grander word?
in English "a kitchen table" is not just the table in the kitchen but suggests the thing around which family resolve issues, is there a similar term?

Answer

chopping board = "tagliere".
dinner table = "tavolo da pranzo" or "tavola". Especially in the expression "a tavola". For instance when someone announces that dinner's ready and the guests are invited to take their seats, he says: "E' pronto. A tavola!".
kitchen table = "tavolo da cucina" (physical). I'm not aware of an Italian expression that translates the extended meaning you've written.

Answer

Tavolo = table, desk.
Tavola = table (as in "multiplication table"), slab, plank, and table (especially the one around which we seat for eating).
Tavolino = little table.