Pre-tax dollars
Money that comes out of gross pay before income tax is worked out, so it never enters taxable income for the year and the saving lands at your top rate.
A pre-tax amount comes out of gross pay, so the year's income tax is worked out on a smaller figure. In the United States the usual examples are a traditional 401(k) deferral, health insurance premiums paid through an employer plan, and contributions to a health savings account or a flexible spending account. The saving is the amount contributed multiplied by your marginal tax rate, because the contribution comes off the top of your income rather than the middle. A contribution big enough to carry you down into a lower bracket saves less than that on the part below the threshold.
Not every pre-tax dollar is pre-tax against the same taxes, and this is the detail that gets missed. A traditional 401(k) deferral escapes federal income tax but still pays Social Security and Medicare tax. Amounts routed through an employer cafeteria plan, such as health premiums and flexible spending account contributions, escape both. Two contributions of identical size can therefore save different amounts.
The mistake is treating the deduction as the whole story. A retirement contribution made with pre-tax dollars is tax deferred rather than exempt, so the tax reappears at withdrawal at whatever rate applies then. Choosing pre-tax over after-tax is a wager that your rate later will be no higher than your rate now. Money spent from an account funded pre-tax, such as a medical bill paid from a health account, is the case where the tax genuinely never arrives.