An Apostolate is the work of an apostle, not only of the first followers of Christ but of all the faithful who carry on the original mission entrusted by the Savior to the twelve to “make disciples of all nations” (Matthew 28:19). The apostolate belongs essentially to the order of grace. Its purpose is not temporal welfare, however noble, but to bring people to the knowledge and love of Christ and, through obedience to his teaching, help them attain life everlasting. (from Catholic Culture)
The lay apostolate is more fully defined in the encyclical of Pope Pius XII “Guiding Principles Of The Lay Apostolate” (1957)