I can't tell you about the implementation of the ban, but I can tell you its source. The Second Lateran Council--Canon 29 to be precise--under Pope Innocent II in 1139 banned the use of crossbows against Christians. The below seems to be the agreed-upon translation:
We prohibit under anathema that murderous art of crossbowmen and archers, which is hateful to God, to be employed against Christians and Catholics from now on.
Here's the entirety of the document in English
Hopefully another of our flaired users will be able to shed some light on the majority of your question.
/u/Domini_Canes has already provided the relevant passage from the Second Lateran council. One thing to point out about that passage is that it is generally interpreted now to mean a ban on both crossbows and longbows not just a ban on crossbows. The use of separate terms for crossbowmen and archers in the text suggests that Innocent II was more concerned with stopping Christians from shooting each other than he was with specifically the crossbow.
I can't comment on the Second Lateran Council's effect on religious institutions but on the secular world it had basically no effect at all. The reason for this becomes pretty apparent when you consider some of other activities condemned by the council. For example:
Canon 14 bans:
We entirely forbid, moreover, those abominable jousts and tournaments in which knights come together by agreement and rashly engage in showing off their physical prowess and daring, and which often result in human deaths and danger to souls.
While the most ludicrous is probably Canon 12 which opens with:
We decree that the truce {11} is to be inviolably observed by all from sunset on Wednesday until sunrise on Monday, and from Advent until the octave of the Epiphany, and from Quinquagesima until the octave of Easter.
This latter text mirrors the words of the Peace of God from the previous century (I can't remember exactly which version of that Peace it was off the top of my head...) so it isn't brand new to this council but restricting warfare to just Monday-Wednesday is asking a lot from medieval nobility.
Jousting, tournaments, crossbows, and warfare on Thursdays-Sundays continued to be fairly commonplace during the twelfth century and later in the Middle Ages despite Innocent's condemnations.