I found the function
$$f_i{(x)}=\frac{\prod_{j=1,j\neq i}^n(x-x_j)}{\prod_{j=1,j\neq i}^n(x_i-x_j)}$$
on "Answer Book for Calculus" by Micheal Spivak -- question 3, 6(a). Which stated that this polinomial equaled $1$ at $x_i$ (which of course, is clear to me) and that $f_i(x)=0$ at $x=x_j$. My problem is with the last statement. Since $j$ is part of the index, how could there be a number $x$ equal to $x_j$ which isn´t a number per say?
I am not sure I got your question right, but maybe my explanation helps anyway.
You have a given pool of numbers, which we call $x_1...x_n$. For any $i \in \{1...n\}$ you define a function $f_i$ with the property that $f_i(x_i) = 1$ and $f_i(x_j) = 0$ for $j \neq i$. When defining $f_i$ the variable $j$ is not in use, so we can use it as an index. Though, when evaluating $f_i(x_j)$, the variable $j$ is in use, so you should use another index like $k$, when plugging $x_j$ in the defining term...