Once upon a time, most American adults were employed, and worked many years for the same company. In the past three decades, so many people are either self-employed or work only a few years before moving to the next company, that the traditional retirement pension was too cumbersome to be effective.
A 401(k) is a retirement savings account, sometimes fed by an employer, but in most cases, the primary contributor is worker/owner of the account. The contributor may begin withdrawing funds at the age of 59 ½ without incurring a tax penalty.
The 403(b) is similar to the 401(k) plan, but is specifically for self-employed ministers, some hospital service organizations, public education organizations, and some non-profit employers (with 501(c)(3) organization).