the pumping length
The pumping length is the threshold beyond which the looping guaranteed by the pumping lemma must kick in. Think of a DFA as a small parking garage with a fixed number of stalls (states). The moment a string is long enough that the car has parked in more stalls than exist, it must have re-parked in one — and that re-park is the loop. The pumping length p is exactly 'long enough' for that to be forced.
Concretely, you can always take p to be the number of states of any DFA that recognizes the language. Reading a string of length at least p means the run passes through at least p+1 states (the start plus one per symbol), and with only p distinct states, two of those must coincide by the pigeonhole principle — guaranteeing a loop within the first p symbols (that is why the lemma's condition |xy| ≤ p holds). Strings SHORTER than p carry no such guarantee; the lemma is silent about them.
Two honest points. First, p depends on the language, not on the string — once fixed, it works for all sufficiently long strings. Second, in a NON-regularity proof you do not get to choose p; the lemma says 'there EXISTS some p', so you must let an adversary hand you an unknown p and then defeat it for that arbitrary value. You only control which long string and which pumping count you use against it.
If a language is recognized by a DFA with 5 states, then p = 5 works: any accepted string of length ≥ 5 forces a repeated state within its first 5 symbols, so it splits as xyz with |xy| ≤ 5 and y pumpable.
Pumping length = number of DFA states is always a safe choice.
p is a sufficient threshold, not a tight one — the minimal pumping length can be smaller than the state count, and the lemma only promises SOME working p, never the smallest. In proofs, treat p as given and unknown.