Dog-Cat Introductions
Contrary to what old cliches might have you believe, dogs and cats can and frequently do live together just fine. As with all else, however, it takes patience, careful management, and respect for both animals' comfort and safety to integrate a foster dog into a household with cats.
Choosing Candidates
The first priority, of course, should be choosing a foster dog who isn't likely to hurt your cat. Dogs who will injure or kill a household cat are actually a very small minority (although rather more of them will hurt a strange cat who wanders onto their home turf); while most dogs are curious about cats, and many will gleefully chase a cat who runs from them, they are generally more inclined to view it as a game than as a serious kill-the-prey hunt. The cat's likely to have a very different perspective on matters, though.
Some cats are dog savvy and/or inherently courageous and will stand their ground when approached by a dog. These cats are likely to be able to handle themselves fine with the foster pup (although you should still give them a safe place as described below, just in case), provided you pick a dog who isn't prone to kittycide.
Other cats, however, are more timid and/or have no experience with dogs (or, worse, have negative experiences of being chased or even injured by dogs previously). If your cat is one of these, you will have to be extra considerate of your cat's stress level, and in extreme cases, perhaps reconsider fostering at all.
A calmer, older dog may be less stressful to your cat than a rambunctious puppy who wants to play (read: chase the cat around) constantly. You might consider fostering a senior dog, at least the first time, to see how your cat does with a relatively unthreatening canine. While I'm of the view that each dog should be evaluated as an individual, not based on breed stereotypes (especially if we're talking about pound puppies!), it nevertheless might be wise to avoid breeds known for high prey drive: sighthounds, terriers, and other dogs bred specifically for hunting smaller animals.
Most shelters evaluate whether a dog is "cat friendly" by putting the dog on a leash and walking it past a cat who is in a cage or a crate. If the dog fixates on the cat with an intense stare and can't be easily distracted, starts to stalk the crate, or actually attacks, that's a fail; however, a pass response is still a little ambiguous, because the test does not account for how the dog will respond to a clearly visible cat who runs away. So even if your new foster pup comes with a stamp saying "cat friendly," don't take too much for granted.
Preparing the Safehouse
Before bringing your foster mutt home, make sure that your cat has a safe refuge where the dog cannot bother her. Separating the animals with baby gates that the cat can jump over, but the dog can't, is one popular technique. (Putting the baby gate at the top of the stairs, if you have that option, can prevent all but the most athletic dogs from jumping over the barrier.) Creating a high perch for the cat on top of a bookshelf, or setting up a big cat tree, are other alternatives. You may want to give the cat an entire safe room and simply close the door to keep the dog out. Wherever you put it, the refuge should have easily accessible food, water, and a litterbox that the cat can use without being disturbed or having to run past the dog to get there.
The cat should have "escape points" (vertically raised perches, such as high cabinets and other furniture, that the dog cannot reach) in every room where he might encounter the foster mutt. Clear off countertops and shelves so that knick-knacks don't get broken by a fleeing feline.
Introducing Fido
Expect introductions to go slowly, and don't push the animals to spend time close to one another immediately (or ever, really). Cats frequently take much longer to adjust to household changes than dogs do. It may be anywhere from 30 days to six months before your kitty stops hiding under the furniture.
Always supervise the animals closely in the beginning. Separate them by crating the dog and/or confining the cat to a different room when you can't be around to keep an eye on them.
Instead of introducing the animals face to face immediately, allow them to investigate each other via scent first. Let the dog sniff the cat's room when the cat is not in there; allow the cat to check out the dog's empty bed by moving the bed into the cat's refuge room for a while. Watch for signs of fear or inappropriate excitement.
When you're ready for them to meet up close and personal, have the dog on a loose, relaxed leash. Praise and reward your foster mutt for any signs of calmness and orienting back toward you/ignoring the cat. Keep such meetings short -- even 30 seconds may be plenty the first time -- and watch both animals carefully to see how they're responding. If they both seem relaxed, you can relax a little more too. If not, continue to proceed at a snail's pace.
Manage the animals ruthlessly to avoid any opportunity for the dog to chase the cat. Even one instance can teach the dog that it is huge fun to chase a cat -- way more fun than any reward you can offer for not doing it -- and can leave the cat terrified for months. Reward the dog for behaving calmly and focusing on you when the cat is within view, and reward the cat too if your cat is inclined to take rewards.
It may only take a few days or weeks before the animals are getting along comfortably. But until you're sure they're the best of friends, always put safety first.
