Suppose that $X$ is an infinite set and $A$ is a unital $C^*$-algebra. The tensor product $\bigotimes_X A$ is defined to be the closed linear span of $\bigotimes_{x\in X }a_x$, where $a_x\in A$ for all $x\in A$ and $a_x=1$ for all but finitely many $x\in X$.
I feel confused about the above definition. Is every element of $\otimes_X A$ the form of $a_1\otimes\cdots\otimes a_n \otimes 1\cdots \otimes 1\otimes\cdots$ for some $n$?
If $X=\mathbb N$, the algebraic tensor product is the span of the elementary tensors $a_1\otimes\cdots\otimes a_n\otimes 1\otimes\cdots$. So sums of the form $$\tag1 \sum_{k=1}^m a_{k1}\otimes\cdots\otimes a_{kn}\otimes 1\otimes\cdots. $$ The C$^*$-tensor product is the closure of the sets of elements of the form $(1)$. The key in defining the tensor product is the norm that you use. Depending on the C$^*$-algebra $A$, there could be more than one possible.
When $X$ is an arbitrary set, you need to write things more carefully (although the spirit is what was said above). A "tuple" indexed by $X$ is a function $f:X\to A$. In this case you can identity the elementary tensors with those maps $f:X\to A$ such that $f(x)=1$ for all by finitely many $x\in X$. Then to define the algbraic tensor product $\bigotimes_X A$ you take the formal span of all those functions and quotient by the subspace given by the tensor relations.