Shabbat presents certain challenges for pet owners, but many, many pet owners are also Shabbat-observant Jews, and becoming familiar with a few basic rules and concepts is all that is needed.
There are four primary areas affected by the Shabbat laws:
3) Walking your pet
4) Touching and petting
Feeding pets on Shabbat is permitted. In fact we are required by the Torah to feed dependant animals every day before we eat ourselves. This is derived from the verse (Deuteronomy 11:15): "I will give grass in your fields for your animal, and you shall eat and be satisfied"--food for the animal is mentioned before food for oneself!
Trapping, defined as any act which restricts the freedom of an animal, is forbidden on Shabbat.
This prohibition is not relevant to many pets, because the prohibition against "trapping" only applies to animals which occasionally escape and are difficult to catch, not to domesticated animals which do not normally run away, since these are considered to be "trapped" already -- due to their nature and/or training.
If your pet is prone to running away, the following rules apply: 1) It is forbidden to put a leash on the pet in a public area, or any place where it would take more than one lunge to catch it; leash the pet before you leave the house. 2) One may not close a window (through which the pet can feasibly escape) or door while the pet is in the home. This problem can be circumvented by securing your pet every time you open and close the door or by bodily blocking the doorway when you open the door so that there is never enough space for the pet to fit through.
Walking a Pet
Walking a pet on Shabbat is not problematic per se, as long as you avoid carrying the animal or any pet accessories in the "public domain" (See The Shabbat Laws).
While the pet may be collared and leashed, it must be clear that you are walking an animal, not carrying a leash. The pet must therefore remain close to you at all times, and the length of the leash should remain taut; never sagging within a handbreadth of the ground, and no more than a handbreadth of extra leash should dangle from your hand.
Touching and Petting
Any object which offers no immediate practical use is called muktzah, and may not be handled on Shabbat. According to conventional Jewish law, animals, too, fall into this category. While household pets may be an exception -- as will be explained shortly -- let us first discuss conventional halachah with regards to handling animals.
It is forbidden to pet, hold, or stroke an animal on Shabbat. An exception to the no-handling rule is if the animal is in pain or discomfort; in such an instance it is permitted to touch it in order to ease its pain. For example, one is allowed to apply oil or an ointment to a wound, or help an animal which is having difficulty walking.
This exception only applies to the rules of muktzah. The laws of muktzah are of rabbinic origin, and were waived by the rabbis in an instance of animal pain or discomfort. However, in a case when the animal's life may be in danger, it is permitted to ask a non-Jew to do any activity which is ordinarily forbidden on Shabbat.
All of the above, as we said, is the conventional law pertaining to handling animals on Shabbat. It has been argued by certain prominent halachic authorities that household pets are not included in the category of muktzah at all, because they have an "immediate practical use" -- namely, providing people with pleasure and companionship. There are others who disagree, maintaining that the rabbinic prohibition against handling animals on Shabbat was imposed across the board. As there are differing opinions in this matter, speak to your rabbi, who will advise you regarding your particular situation.
[Animal litter is also muktzah. But you may clean it up if it is in your home and disturbing you.]
Note: All these rules apply to major Jewish holidays too, with the exception of the rules regarding carrying in the Public Domain, which are not applicable on Jewish holidays.