53
Deterministic Finite Automatons (DFAs) and Nondeterministic Finite Automatons (NFAs) have exactly the same capabilities and limitations. The only difference is notational convenience.
A finite automaton is a processor that has states and reads input, each input character potentially setting it into another state. For example, a state might be "just read two Cs in a row" or "am starting a word". These are usually used for quick scans of text to find patterns, such as lexical scanning of source code to turn it into tokens.
A deterministic finite automaton is in one state at a time, which is implementable. A nondeterministic finite automaton can be in more than one state at a time: for example, in a language where identifiers can begin with a digit, there might be a state "reading a number" and another state "reading an identifier", and an NFA could be in both at the same time when reading something starting "123". Which state actually applies would depend on whether it encountered something not numeric before the end of the word.
Now, we can express "reading a number or identifier" as a state itself, and suddenly we don't need the NFA. If we express combinations of states in an NFA as states themselves, we've got a DFA with a lot more states than the NFA, but which does the same thing.
It's a matter of which is easier to read or write or deal with. DFAs are easier to understand per se, but NFAs are generally smaller.
Here's a non-technical answer from Microsoft:
DFA engines run in linear time because they do not require backtracking (and thus they never test the same character twice). They can also guarantee matching the longest possible string. However, since a DFA engine contains only finite state, it cannot match a pattern with backreferences, and because it does not construct an explicit expansion, it cannot capture subexpressions.
Traditional NFA engines run so-called "greedy" match backtracking algorithms, testing all possible expansions of a regular expression in a specific order and accepting the first match. Because a traditional NFA constructs a specific expansion of the regular expression for a successful match, it can capture subexpression matches and matching backreferences. However, because a traditional NFA backtracks, it can visit exactly the same state multiple times if the state is arrived at over different paths. As a result, it can run exponentially slowly in the worst case. Because a traditional NFA accepts the first match it finds, it can also leave other (possibly longer) matches undiscovered.
POSIX NFA engines are like traditional NFA engines, except that they continue to backtrack until they can guarantee that they have found the longest match possible. As a result, a POSIX NFA engine is slower than a traditional NFA engine, and when using a POSIX NFA you cannot favor a shorter match over a longer one by changing the order of the backtracking search.
Traditional NFA engines are favored by programmers because they are more expressive than either DFA or POSIX NFA engines. Although in the worst case they can run slowly, you can steer them to find matches in linear or polynomial time using patterns that reduce ambiguities and limit backtracking.
[http://msdn.microsoft.com/en-us/library/0yzc2yb0.aspx]