Meaning of ε in NFA-ε?
An epsilon transition is simply a transition that doesn't consume any characters in the input string. 

Taking it is optional, and it is what makes the non-deterministic finite automata non-deterministic: by allowing it to 'branch' it can 'explore' two plausible paths through the language it accepts. It is essentially just a space saving measure: DFAs (which do not permit epsilon transitions) are computationally equivalent to NFAs. There are steps to convert an NFA to an equivalent DFA, removing epsilon transitions is part of this conversion. 

To explicitly draw out answers to your questions: In your example it is akin to a 'NO_INPUT' symbol, but more like 'NO_COST'. Think of it like this: Each edge has a 'cost' in the symbol along it that you must pay to transition along that edge. An epsilon has no cost. You don't 'spend' any characters from your input string to traverse that edge. Because of that, you now have multiple paths: Take the epsilon transition (or transitions) and spend no symbols from the input string, or 'spend' an input string along an edge. So it essentially the computation 'branches', and your NFA accepts a string if any path through it concludes in a success state.