guessing and verifying
Here is a wonderfully useful way to read an NFA: pretend the machine has a magic oracle that, at the crucial moment, GUESSES the right move — and then the rest of the run just VERIFIES that the guess was good. If a correct guess exists, acceptance follows; if no guess could ever work, the string is rejected. This guess-then-verify mindset turns many tangled design problems into one-line descriptions.
Concretely, suppose you want to accept strings that contain the block aba somewhere. You design the NFA to sit in a waiting state, and at every position it may either keep waiting or GUESS "the aba I am promised begins right here." Having guessed, it commits to checking a, then b, then a; if those three symbols really are present it reaches an accept state. The guess is not magic foresight — it is a branch in the tree of possibilities, and the existential acceptance rule means we only need one branch where the guess turns out to be right.
This is the same intuition that, scaled up, defines the class NP: a problem is in NP when a proposed solution (a guess, or certificate) can be VERIFIED quickly even if finding it seems hard. For finite automata the guessing is always cheap to simulate away — the subset construction removes it entirely. The deep open question P versus NP asks whether that is also true for those harder problems. So this small idea is a first taste of one of the biggest questions in computer science.
"The third symbol from the end is a": instead of remembering everything, the NFA waits in a start state, GUESSES at some position that "this is the third-from-last a," reads a, then any symbol, then any symbol, and lands on accept exactly when the guessed position really was third from the end. Verification (read three more) confirms the guess.
Design by guessing the answer, then writing the machine that verifies the guess.
The guess is a metaphor for branching, not a real algorithm step. For NFAs it is always removable; whether it is removable cheaply for NP problems is the open P-versus-NP question.