Judaism places great stress on proper treatment of animals. Unnecessary cruelty to animals is strictly forbidden, and in many cases, animals are accorded the same sensitivity as human beings. This concern for the welfare of animals is unique to Judaism; Christianity does not share this value, nor did most civilized nations until quite recently. Cruelty to animals was not outlawed until the 1800s.
Judaism expresses no definitive opinion as to whether animals are capable of experiencing physical or psychological pain as humans do; however, Judaism has always recognized the link between the way a person treats animals and the way a person treats human beings. A person who is cruel to a defenseless animal will undoubtedly be cruel to defenseless people, and a person who cares for the lowest of creatures will certainly care for his fellow man.
Jacob, Moses and David were all shepherds, people who cared for animals. The Talmud specifically states that Moses was chosen for his mission because of his skill in caring for animals. The Holy One, Blessed Be He, said Since you are merciful to the flock of a human being, you shall be the shepherd of My flock, Israel. Likewise Rebekah was chosen as a wife for Isaac because of her kindness to animals. When Abraham's servant asked for water for himself, she volunteered to water his camels as well, and thereby proved herself a worthy wife.
On the other hand, the two hunters in the Bible, Nimrod and Esau, are both depicted as villains. A great rabbi who was insensitive to the fear of a calf being led to slaughter was punished with years of pain.
In the Torah, humanity is given dominion over animals, and has the right to use animals for legitimate needs. Animal flesh can be consumed for food; animal skins can be used for clothing; the Torah itself must be written on parchment, that is, animal hides.
However, we are permitted to use animals in this way only when there is a genuine, legitimate need, and we must do so in the manner that causes the animal the least suffering. Kosher slaughtering is designed to be as fast and painless as possible, and if anything occurs that might cause pain (such as a nick in the slaughtering knife or a delay in the cutting), the flesh may not be consumed. Hunting for sport is strictly prohibited, and hunting and trapping for legitimate needs is permissible only when it is done in the least painful way possible.
The laws regarding treatment of animals are referred to as Tzar Baalei Chayim, prevention of cruelty to animals.
Under Jewish law, animals have some of the same rights as humans do. Animals rest on Shabbat, as humans do. We are forbidden to muzzle an ox while it is working in the field, just as we must allow human workers to eat from the produce they are harvesting.
Several commandments demonstrate concern for the physical or psychological suffering of animals. We may not plow a field using animals of different species, because this would be a hardship to the animals. We are required to relieve an animal of its burden, even if we do not know its owner, or even if it is ownerless. We are not permitted to kill an animal in the same day as its young, and are specifically commanded to send away a mother bird when taking the eggs, because of the psychological distress this would cause the animal. In fact, the Torah specifically says that a person who sends away the mother bird will be rewarded with long life, precisely the same reward that is given for honoring mother and father. This should give some indication of the importance of this law.
We are permitted to violate the Sabbath to some extent to rescue an animal in pain or at risk of death.
Jewish law does not prohibit keeping pets, and indeed many observant Jews have dogs, cats or other household pets.
As with all animals, we are required to feed our pets before ourselves, and make arrangements for feeding our pets before we obtain them. Also, like all animals, household pets are entitled to Sabbath rest, thus you cannot have your dog retrieve the paper for you on Shabbat, etc.
As far as I have been able to ascertain, it is permissible to feed non-kosher food to pets. As I understand it, it is permissible to use products of non-kosher animals as long as you don't eat them; for example, it is permissible to use a toothpaste that contains non-kosher ingredients as long as the toothpaste is not fit for human consumption. Likewise, it is permissible to feed non-kosher food to your pets, as long as you do not consume it yourself.
The laws of Passover, however, are somewhat broader. During Passover, it is impermissible to have any chametz (leavened grain products) in your home, or to derive any benefit from chametz, thus you cannot use chametz to feed your pets. You must either feed your pet something that contains no chametz (such as 100% beef dog food, kosher for Passover table scraps, or matzah meal to feed fish or rodents) or temporarily sell the pets to a non-Jew, as you temporarily sell your pots and pans to a non-Jew during the holiday.
It is a violation of Jewish law to neuter a pet. The Torah prohibits castrating males of any species. Although this law does not apply to neutering female pets, neutering of females is prohibited by general laws relating to unnecessary cruelty to animals.
It is a violation of the general prohibition against cruelty to animals to have your pet physically altered in any way without a genuine, legitimate need. For example, declawing cats and docking the ears or tails of dogs are forbidden.
Sources: Judaism 101