to loop forever

To loop forever is for the machine to keep running and never stop, never reaching a verdict. This is the third outcome, the one that has no counterpart for a DFA. A DFA always reads its finite input and stops; a Turing machine, free to move its head either way and rewrite cells endlessly, can get caught in motion that never ends, like a person told to keep walking right along an infinite road with no instruction to ever turn back.

Looping does not require literally repeating the same configuration (though that is one way it happens, since once a configuration recurs the machine is doomed to cycle). It just means the computation is an infinite sequence of configurations: the machine never enters q_accept or q_reject. The looping can be subtle, for example a head that drifts steadily rightward across blanks forever, writing as it goes, never repeating a configuration yet never halting. From the outside you simply see it running, and running, with no way to tell from watching whether it will eventually stop.

Looping is precisely what separates recognizing a language from deciding it. A machine that recognizes a language is allowed to loop forever on inputs that are NOT in the language, as long as it accepts every input that IS in the language. A decider is forbidden from looping: it must halt on every input. So 'loop forever' is not a bug to be wished away; it is a built-in possibility of the model, and the impossibility of detecting it in general (the halting problem) is one of the deepest results in computer science.