The debate over eating meat and vegetarianism is ongoing. In Jewish writings, both views do receive recognition. On the one hand, Hashem gave us the ability to conquer the land and all that is in it. This allows us to use animals to our benefit, provided we do not violate צער בעלי חיים, mistreating animals.
On the other hand, some view שחיטה, ritual slaughtering, as possessing a certain element of cruelty. I was told by Rabbi Eliezer Wolff, a שוחט of עופות, that there exists an interesting Halacha conforming this idea.
We are permitted to do שחיטה on all of the Yom Tovim except Rosh Hashanah. At the time of the Talmud, there was no refrigeration, so the only way to have fresh meat was to slaughter during the Chag. The Rabbis felt that despite this allowance on other holidays, it was not appropriate for Rosh Hashanah. This was a day when we ask Hashem for mercy and שחיטה still represented an element of cruelty. Therefore, the שחיטה allowance did not extend to Rosh Hashanah.