What's meaning of the inverted Greek letter iota “ι” in Principia Mathematica I* 14?

Question

The inverted iota always followed by a variable as following: (ιx)ϕx Some book say the formula should be read "The x such that x is ϕ". Does it is meaning as following: x=ϕ assign ϕ to variable x?

Answer 1

It is not an assignment. $\phi$ is a predicate. $(\iota x)\phi x$ is the one and only object $x$ for which the predicate $\phi$ is true.

For example, if $\phi x$ means "$x$ is the current monarch of the United Kingdom" then $(\iota x)\phi x$ denotes King Charles III. As another example, $(\iota x). x\ge 0\mathrel . x^2=3$ denotes the number $\sqrt 3$.

This is explained in the front matter of the Principia, including what the meaning is when there is no unique $x$ (either because there is none or because there is more than one). If the denoted predicate is not unique, then the truth value of the formula is false.

Answer 2

This is the descriptor operator. $(\iota x)\varphi x $ is the unique $x $ with the property specified by $\varphi $ (should it be the case that, indeed, there is precisely one such $x $). The Wikipedia entry on Principia has a very decent explanation of their notation.