Difference between reductions in algebraic problems versus reductions in computational intractability
Reductions in computational complexity are similar to the reduction you described, but are usually given with bounds on the time or space. To be useful, these bounds are usually inconsequential (ie. the same or smaller) to the bounds of solving the problem.

Reductions in computational complexity are about converting one problem to another equivalent one. Usually one would also give bounds on how hard (how much time/space) a reduction takes as well. For example, all NP-complete problems have reductions to each-other, and their reductions are polynomially bounded. Since algorithms solving NP-complete problems are suspected to take exponential time on a deterministic Turing machine, the polynomial time reductions are inconsequential (compared with the time of actually solving the reduced problem), and thus these reductions make all of the NP-complete problems essentially equivalent. There can reductions between problems in other classes of complexity as well, and to be useful they would likely be bounded by something considered inconsequential to solving them.