a multitape Turing machine
Picture working a hard problem at a desk. With one sheet of paper you constantly shuffle back and forth: scratch work here, the running total there, the original numbers up top, erasing and rewriting as you go. Give yourself three separate sheets instead, one for each job, and the work feels easier because you never have to skip past one thing to reach another. A multitape Turing machine is exactly this: a Turing machine with several tapes instead of one, each with its own read-write head that can move independently.
Formally, a k-tape Turing machine has k tapes and k heads. In one step it reads the k symbols currently under its heads, and based on its state and those k symbols it writes a new symbol on each tape, moves each head left or right (or stays put) independently, and changes state. So a single transition now depends on a whole tuple of symbols at once: δ(q, a1, ..., ak) = (p, b1, ..., bk, D1, ..., Dk), where δ (delta) is the transition function and each Di is a direction. Input usually starts on tape 1 with the others blank. The extra tapes are just extra scratch space and extra heads, nothing more exotic.
The point of this variant is convenience, not power. It is far easier to design a 3-tape machine (one tape for input, one to count, one for output) than to cram everything onto one tape. The remarkable fact, proved by simulation, is that any k-tape machine can be mimicked by an ordinary single-tape machine: lay the k tapes side by side as k tracks on one tape, mark each head position, and sweep across to gather all k symbols before each step. They recognize exactly the same languages. The single-tape simulation is slower by only a polynomial factor (roughly squaring the time), which is why complexity theory can speak of polynomial time without naming the number of tapes.
To decide {ww : w in {0,1}*} a 2-tape machine copies the first half onto tape 2, then walks both heads forward in lockstep comparing symbols. On one tape the same job means tediously bouncing back and forth marking matched positions; the second tape removes all that bouncing.
Two tapes make a comparison task trivial; one tape can still do it, just with more head-shuffling.
Extra tapes never let a Turing machine recognize a new language; they only save effort and (at most polynomially) time. Believing multitape machines are strictly more powerful is a common beginner mistake.