A non-deterministic Turing machine (NTM) differs in that, rather than a single instruction triplet, the transition rule may specify a number of alternate instructions. At each step of the computation we can imagine that the computer "branches" into many copies, each of which executes one of the possible instructions. Whereas a DTM has a single "computation path" that it follows, a NTM has a "computation tree". If any branch of the tree halts with an "accept" condition, we say that the NTM accepts the input.
Intuitively it seems that the NTM is more powerful than the DTM, since it can execute many possible computations in parallel, requiring only that one of them succeeds. Any computation carried out by a NTM can be simulated by a DTM, although it may require significantly longer time. How much longer is not known in general - this is, in a nutshell, the definition of the "Is P = NP?" problem (see Complexity classes P and NP).