What is a lay apostolate?

I've heard so many Catholics on the Journey Home refer to their apostolate, I thought I'd better look it up.  Well, for starters, it isn't what you do for your individual church or congregation, like parish council or teaching classes.  It's in the secular world, working for the Lord.

A key part of lay apostolate is that it happens in a secular environment, not in church. Vatican Council II's Constitution on the Church spoke of it as a “special vocation” — making faith “present and fruitful” in those places where that can only be done by the laity. What places might those be? The home, the neighborhood, and the workplace come to mind. If Christianity is to be lived out there, it's up to lay people to do it.

Lay apostolate comes in two broad varieties — individual and group. A group apostolate might involve something like running a pregnancy counseling program or operating a values-oriented private school. http://catholicexchange.com/whats-a-lay-apostolate

I encourage you to read the    

DECREE ON THE APOSTOLATE OF THE LAITY
APOSTOLICAM ACTUOSITATEM
SOLEMNLY PROMULGATED BY HIS HOLINESS,
POPE PAUL VI
ON NOVEMBER 18, 1965 

It's really quite inspiring; succinct, yet readable in its thoroughness.
  http://www.vatican.va/archive/hist_councils/ii_vatican_council/documents/vat-ii_decree_19651118_apostolicam-actuositatem_en.html

Our own times require of the laity no less zeal: in fact, modern conditions demand that their apostolate be broadened and intensified. With a constantly increasing population, continual progress in science and technology, and closer interpersonal relationships, the areas for the lay apostolate have been immensely widened particularly in fields that have been for the most part open to the laity alone. These factors have also occasioned new problems which demand their expert attention and study. This apostolate becomes more imperative in view of the fact that many areas of human life have become increasingly autonomous. This is as it should be, but it sometimes involves a degree of departure from the ethical and religious order and a serious danger to Christian life. Besides, in many places where priests are very few or, in some instances, deprived of due freedom for priestly work, the Church could scarcely exist and function without the activity of the laity. . . "

". . . The greatest commandment in the law is to love God with one's whole heart and one's neighbor as oneself (cf. Matt. 22:37-40). Christ made this commandment of love of neighbor His own and enriched it with a new meaning. For He wanted to equate Himself with His brethren as the object of this love when He said, "As long as you did it for one of these, the least of My brethren, you did it for Me" (Matt. 25:40). Assuming human nature, He bound the whole human race to Himself as a family through a certain supernatural solidarity and established charity as the mark of His disciples, saying, "By this will all men know that you are My disciples, if you have love for one another" (John 13:35). . ."

" . . . In the manner of the men and women who helped Paul in spreading the Gospel (cf. Acts 18:18, 26; Rom. 16:3) the laity with the right apostolic attitude supply what is lacking to their brethren and refresh the spirit of pastors and of the rest of the faithful (cf. 1 Cor. 16:17-18). Strengthened by active participation in the liturgical life of their community, they are eager to do their share of the apostolic works of that community. They bring to the Church people who perhaps are far removed from it, earnestly cooperate in presenting the word of God especially by means of catechetical instruction, and offer their special skills to make the care of souls and the administration of the temporalities of the Church more efficient and effective."