Given an automaton and an alphabet $\{a, b\} $,  and the language accepted by the automaton is $ab^*$. Such an automata can be found here:My question is: this automaton cannot process the word $ba$ for example. In general,  do automata have to be able to process any word?
Automata with no transition for some character
terminology;automata;finite automata
In a deterministic automaton, every state must have exactly one transition for each symbol in the alphabet. In a nondeterministic automaton, this requirement is dropped and there could be multiple transitions or no transition for some symbol. If there is no transition for the next character in the string, the automaton rejects its input. This is what happens with the automaton on the left: it rejects the string $ba$ (and every other string beginning with $b$) because it has nowhere to go when it reads that first $b$.