I remember a time, not so long ago, when all the women in church would wear head coverings. Where did this custom come from and why don’t we see it in church today?

The custom and discipline of women wearing head coverings or “mantillas” is rooted in the norm St. Paul prescribed in 1 Corinthians 11:2-6 which says “every woman who prays or prophesies with her head uncovered dishonors her head.
 
In short, this practice is no longer binding to women. Here’s a little background on the matter. In Canon 1262 section 2 of the 1917 Code of Canon Law, the Church maintained the Pauline discipline: Men, in a church or outside a church, while they are assisting at sacred rites, shall be bare-headed, unless the approved mores of the people or peculiar circumstances of things determine otherwise; women, however, shall have a covered head and be modestly dressed, especially when they approach the table of the Lord.” However Canon 6 section 1.1 of the 1983 Code abolished the provisions of the 1917 Code. What this means is that any 1917 Canon that was not somehow incorporated into the 1983 Code is no longer in effect. The 1983 Code did not reissue the canon on women’s head coverings, making it clear that women no longer have any canonical obligation to wear a head covering
 
Prior to Vatican II, in the early 1960s, women in the Church began to participate at Mass without head coverings. By the early to middle 1970s, very few women were wearing head coverings at Mass. While the Church never made a canonical pronouncement, canon law does allow for modification of a custom, provided that the Church permits the modification. In summary, the custom began to change among the faithful, and the Church deemed it prudent to let the former custom naturally pass away. Consequently, the 1983 Code formally abolished the custom of head covering for women because they ceased to be operative for a good number of years. While no longer required, head coverings have always been a matter of culture and piety. Head coverings for women were a way of showing respect and reverence to God and Jesus Christ who is our true Head and Shepherd. This is why it is polite for a man to remove his hat while in church...out of reverence and respect for Christ who is our true Head and Shepherd. The question of whether to wear a head covering is left up to the individual.