I can't answer for European traditions of animal sacrifice, but in the case of Jewish sacrifice, the practice largely ended with the destruction of the Temple in 70CE. Over time the practice of sacrifice had become highly centralized at the Temple in Jerusalem, so once it was destroyed the practice simply ended. Moreover, once centuries had past, the "institutional knowledge" for performing sacrifices had largely been lost too.

Animal sacrifice still has an important legacy in Jewish ritual--the liturgy's structure is loosely based on the sacrificial practice, and certain parts of the liturgy discuss it. The Musaf service, which is an extra service for the Sabbath and holidays, in emulation of the extra sacrificial service for those days, talks about the practice. There's also something called the Avodah service on Yom Kippur, where the sacrificial service for the day is basically re-told completely in liturgical form.

So in summary, sacrifice ended in 70CE, and it ended because the Temple was destroyed. It was a matter of practicability--it was no longer possible to perform sacrifices as they had been, not a decision by anyone in particular in the Jewish community.