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A 403(b) is a retirement plan provided by employers in nonprofit organizations. This includes school systems, hospitals and healthcare systems and religious organizations. 403(b) plans are similar to 401(k) plans which are provided by employers engaged in for-profit businesses. 

Contributions to a 403(b) plan are pre-tax and come directly out of your paycheck. In the eyes of the government you've actually earned less (your 403(b) contribution is deducted from your earnings) so you are taxed less.

Here's an example:  A $100 contribution to a 403(b) reduces federal income taxes by roughly $28 (assuming you are in the 28% tax bracket). In effect, your $100 contribution costs you only $72.