What is a fixed point in the context of Roger's fixed-point theorem?
The notation is explained in the Wikipedia entry (though regrettably after its first use): for partial functions $f,g$, we say that $f\simeq g$ if for all inputs $x$, $f$ halts on $x$ iff $g$ halts on $x$, and if both halt on an input $x$, then $f(x) = g(x)$. In other words, $f$ and $g$ compute the same partial function, and so they are semantically equivalent: they result in the same outcome.

The idea behind semantic equivalence is that two programs might be different but could be equivalent in the sense that they compute the same function. For example, if there are two statements $x \gets 0; y \gets 1$ and we switch their order to $y \gets 1; x \gets 0$ then the resulting programs are different syntactically but equivalent semantically.

Roger's theorem shows that for every total computable function $P$, which we think of as a transformation rule for programs, there is some program $e$ that is equivalent to $P(e)$. The rest of the Wikipedia entry explains why this is useful.