In what sense are independent random vectors "independent"?

Question

Let's say we have independent random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ and $\boldsymbol{Y}=(Y_{1},...,Y_{q})$. What is it that makes them independent? Are the $X_i$ independent of the $X_j$ ($i \neq j$)? Are the $X_i$ independent of the $Y_j$? Is it that $\text{Cov}(X_i,X_j)=0, \forall i\neq j$, or $\text{Cov}(X_i,Y_j)=0$? Are these vectors independent in some other sense?