403(b) are basically just 401(k)s for non-profit organizations.
A 403(b) Plan is essentially a 401(k) for publicly-funded institutions such as public schools and universities, and certain hospitals, and non-profit organizations. They are sometimes called TSAs, short for Tax-Sheltered Annuity, but this is outdated, and a misnomer since they do not need to use annuity products.
The contributions are deducted from the paychecks in the same manner they would be for a 401(k), and the assets grow tax-deferred within the account. A Roth 403(b) is uncommon but sometimes offered.
One of the only differences is the the 15-year service based catch-up, which, if the employee meets certain criteria and has been with the employer for over 15 years, allows him or her to contribute an additional $3,000 a year, in addition to the regular contribution limits and catch-up provisions that exist in a 401(k). Lifetime maximum of that additional catch-up is capped at $15,000.