the tape

The tape is the Turing machine's notebook: a single long strip divided into equal cells, stretching away forever in at least one direction. Each cell holds exactly one symbol at a time. Unlike a finite automaton, which can only remember which of finitely many states it is in, or a pushdown automaton, which can only see the top of its stack, the machine can use the tape as freely as you use scratch paper: read any cell, erase it, write something new, come back to it later.

Two features make the tape powerful. First, it is unbounded: there is no limit set in advance on how many cells the machine may use, so it never runs out of room. (It is usually called 'infinite', but at any moment only finitely many cells differ from blank, because the machine has taken only finitely many steps.) Second, it is read-and-write: the machine can both inspect a cell and overwrite it, any number of times. The input string is written on the tape at the start, one symbol per cell, with the blank symbol filling every other cell; that same tape then doubles as working memory and, if the machine computes a function, as the place the output is left.

The unbounded read-write tape is the single ingredient that lifts the Turing machine above all weaker models. A stack lets a pushdown automaton remember nested structure but forces it to forget the bottom to see the top; the tape has no such discipline. Honest caveat: 'unbounded' does not mean the machine has actually-infinite memory sitting around. It means there is no fixed ceiling. For any one halting computation the machine touches only finitely many cells, just as a real program uses only as much memory as it happens to need.