A few years ago the fast answer and the thinking answer came from separate models. Today it is typically one setting on the same model: low, medium or high. A higher value means the model takes more intermediate steps before it speaks — and you pay for those steps as well.
The difference is not small. On complex, multi-step tasks — a derivation, hunting a bug, reconciling several conditions — the higher setting is measurably more accurate. On a simple task it is only slower and dearer: the same answer, by a longer road.
The most common mistake is that people set it to one level and leave it there. Correct use is task-dependent: low for the bulk, simple operations, high for the rare, hard decisions. Set everything to high and you pay the larger part of your bill for steps that were never needed.
What happens at each level and what you pay for it is set out in Reasoning Mode: What Do You Get When the Model Thinks Longer?, a separate article. Directly related terms: chain-of-thought, token and inference.