Need explanations on families of sets

Question

I am currently reading Naive Set Theory by P. R. Halmos and am quite stumped with the notation used to define a family of sets. E.g, given "a family $\{x_{i}\}$ in $X$". What exactly does this mean? What is $\{x_{i}\}$? Intuitively, I came to the conclusion that $\{x_{i}\}$ is a set of all the elements in $X$ indexed by some set $I$. Am I correct? And, if I am, wouldn't the range of the family function given in the example be equal to $\{x_{i}\}$? Also, is there any standard way to denote a family as a lowercase roman letter, similar to other function denotations?