the language of a Turing machine
Every machine, by its behavior, carves out a set of strings: the ones it says yes to. The language of a Turing machine is exactly that set: all the input strings the machine accepts. Just as a DFA's language is the strings it ends in an accept state on, a Turing machine's language is the strings on which it eventually halts in q_accept. It is the machine's 'membership policy' made concrete.
Formally, the language of a Turing machine M, written L(M), is the set of all strings w over the input alphabet such that M accepts w (the computation of M on w halts in q_accept). The crucial subtlety is what happens to strings NOT in L(M): the machine might halt-and-reject them, or it might loop forever on them. Both behaviors are consistent with the same L(M), because L(M) is defined only by which strings are accepted, and says nothing about whether non-accepted strings are rejected or looped on. A language equals L(M) for some machine exactly when it is Turing-recognizable.
L(M) is the link between an individual machine and the abstract languages the theory studies. If M happens to be a decider (always halts), then for the strings not in L(M) the machine definitely rejects, and L(M) is decidable; if M is merely a recognizer, L(M) is recognizable but maybe not decidable. The same language can be the L(M) of many different machines, fast or slow, decider or not, which is why properties of L(M) (what it contains) and properties of M (how many states it has) are very different things, a distinction at the heart of Rice's theorem.
If M accepts a string of as exactly when its length is a power of 2, halting-accepting on a, aa, aaaa, ..., then L(M) = {a^(2^k) : k >= 0}. Whether M rejects or loops on, say, aaa does not change L(M); L(M) is fixed solely by the accepted strings.
L(M) = the set of strings M accepts, regardless of what it does on the rest.
L(M) is defined by acceptance only. Two machines with the same L(M) may behave totally differently on non-members (one rejects, one loops). A set is some L(M) exactly when it is Turing-recognizable.