a configuration (instantaneous description)
A configuration is a complete snapshot of the machine at one instant: a photograph that captures everything you would need to resume the computation if you walked away and came back. For a Turing machine that means three things together: which state the control is in, exactly what is written on the whole tape, and where the head is parked. Nothing else matters, because the machine's next move depends only on these.
A standard, compact way to write a configuration is to write out the tape contents and slot the current state right before the symbol the head is reading. So the string 1 0 q3 1 1 means: the tape holds 1 0 1 1, the machine is in state q3, and the head is on the third cell (the 1 just after q3). Everything to the left of q3 is tape the head has passed; everything from the marked symbol rightward is what remains. Trailing blanks are usually omitted since they are implied. Two configurations are the same exactly when state, tape contents, and head position all match.
Configurations are the vocabulary in which a whole computation is described: the machine moves from one configuration to the next by a single application of the transition function, and a computation is just a chain of configurations. They are also the workhorse of proofs about Turing machines: a starting configuration (state q0, head on the leftmost input symbol), accepting configurations (state q_accept), and rejecting configurations (state q_reject) are special cases, and many arguments work by reasoning about how configurations can follow one another.
Suppose the tape is X a b and the head is on the b in state q5. The configuration is X a q5 b. After δ(q5, b) = (q6, Y, L) the head moves left over the a, giving the next configuration X q6 a Y.
Writing the state just before the scanned cell captures state, tape, and head in one line.
A configuration is the Turing-machine analogue of a pushdown automaton's instantaneous description, but it must record the entire tape (and head position), not just a stack, because any cell may be revisited.