What is Group in Diffie-Hellman?
However, in some protocols descriptions, like TLS 1.3, I find them say
  the client sends "hello message includes Diffie-Hellman public values
  for the client's preferred groups". What is "group" refers to??


The term group is a mathematical concept that guarantees that a specific operation doesn't leave a set, that it is associative, that there is a neutral element and that there's an inverse element for each element.

Diffie-Hellman generally operates in such groups.
Just view it like this: Your set is defined by $p$, e.g. for all integers $x$ in the group $0\leq x &lt;p$. You then take an element $g$ from it and apply the group operation (multiplication modulo $p$) a certain number of times to it (whatever your exponent tells you).

Now every choice of $p$ describes its own group and additionally you can choose groups with an entirely different structure such as the points of elliptic curves over prime fields (which makes this ECDH)