an epsilon-NFA
/ epsilon = EP-si-lon (the symbol ε) /
An epsilon-NFA is a nondeterministic finite automaton that, on top of branching, is also allowed to make free moves on the empty string ε (epsilon). It is the most relaxed of the finite-automaton models: in any state it may guess among several next states, may have no move at all, and may also drift between states without reading input. It is built for designer convenience above all.
Formally it is a five-tuple (Q, Σ, δ, q0, F) just like an NFA, except the transition function is defined on Σ ∪ {ε} — that is, on every real symbol AND on the empty string. So δ(q, a) is a set of states for each symbol a, and δ(q, ε) is the set of states reachable for free. To run it you lean on the epsilon-closure: track the set of possible states, and after every step expand that set by all states reachable through epsilon-moves.
Crucially, the epsilon-NFA is no more powerful than a plain NFA or a DFA. There is a standard procedure to eliminate epsilon-moves (rewire each epsilon-jump by composing it with the real moves on either side, using epsilon-closures), turning an epsilon-NFA into an ordinary NFA recognizing the same language; the subset construction then yields a DFA. This is the third leg of the great equivalence: DFA, NFA, and epsilon-NFA all recognize EXACTLY the regular languages. The epsilon-NFA shines in Thompson's construction, where regular-expression operators map onto tiny epsilon-glued gadgets.
An epsilon-NFA for a* (zero or more a's): a single state s that is both start and accept, with δ(s, a) = {s}, and that is it. The empty string is accepted because s is already accepting via its own epsilon-closure; aaa is accepted by looping. Glue such gadgets with epsilon-moves and you can build a machine for any regular expression.
Epsilon-moves make machines easy to compose; eliminating them recovers an ordinary NFA.
Despite the extra freedom, an epsilon-NFA recognizes only regular languages — the same class as DFAs. The freedom is descriptive, not extra computational power.