This standard definition of pumping lemma from Wikipedia.
Let $L$ be a regular language. Then there exists an integer $p\ge 1$ (depending only on $L$) such that every string $w$ in $L$ of length at least $p$ ($p$ is called the "pumping length") can be written as $w = xyz$ (i.e., $w$ can be divided into three substrings), satisfying the following conditions:
- $|y| \ge 1$
- $|xy| \le p$ and
- for all $i \ge 0$, $xy^iz \in L$.
$y$ is the substring that can be pumped (removed or repeated any number of times, and the resulting string is always in $L$).
What confuses me about the definition of pumping lemma are two requirements: $|y| \ge 1$ and $i \ge 0$, $xy^iz$. The way I read it, that we are required to have $y$ length be equal to one or great, and at the same time, we can completely skip it, since $i \ge 0$, i.e. effectively $|y| = 0 $. Intuitively, it makes sense that we should be able to skip $y$ and still have string be in $L$.