In linear algebra, a linear combination of vectors $v_1,v_2,\ldots, v_n$ is anything of the form $a_1v_1+\ldots +a_nv_n$, where $a_1,\ldots , a_n$ are scalars (referred to as the coefficients of the linear combination).
A set of vectors $\{v_1,\ldots, v_n\}$ is called linearly independent if $a_1v_1+\ldots +a_nv_n=0 \Rightarrow a_1,\ldots ,a_n=0.$
So, the connection between these definitions is that, for a set of linearly independent vectors, the only linear combination of $v_1,\ldots,v_n$ which equals $0$ is when all of the coefficients are $0$. If there is some linear combination of $v_1,\ldots,v_n$ equal to $0$ for which not all of the coefficients are $0$, then $v_1,\ldots,v_n$ are not linearly independent.