The definition of tori – topology

The question may seem silly, but I do not find the answer.

For me, the $n-$torus is “an sphere with $n$ holes”.

Topologically, this is called the connected sum of $n$ tori, is it right?

What about the $T^n=overbrace{S^1times cdots times S^1}^{n ,times}$, what is its name and what are its properties?

Do the two surfaces have some relation?

Thank you so much.