A Basenji is the breed of dog that can't bark. Producing, rather, a yodel-like sound, the Basenji very seldom lets out a single 'woof'.

http://www.breederretriever.com/photopost/data/556/medium/basenji.jpg

Basenji dogs are small and elegant, weigh, on average, between twenty to twenty-four pounds and are, approximately, seventeen inches in height. Originating from Africa, the Basenji, sometimes referred to as the African Barkless Dog, is one of the oldest breeds of dogs, even depicted in the tombs of Egyptian pharaohs, and is known to be intelligent, fast on their feet, courageous and, sometimes, a little mischievous. Similar, in appearance, to a miniature deer the Basenji is a shorthaired dog, varying in colour, with pointed ears and a tightly curled tail. The Basenji has a life span of twelve to fourteen years.

http://www.dogbreedinfo.com/images18/BasenjiPerrinCamping42.JPG

Check out the following site for more info:
http://www.dogbreedinfo.com/basenji.htm

Check out a (Retired Racing) Greyhound.

They are super walking companions and are known as non barking dogs (most are the quiet type). Females range from a medium sized 50lbs to 70lbs, males can be larger up to 75+lbs, but the dogs are not what I would describe as big dogs in that they have a very slight build, they are light on their feet and easy on the house, they curl up into a tiny ball to sleep etc.

Greyhounds have been trained to respect the leash and heal very well, right next to you. They get home after the walk and sleep. Unlike some other breeds mentioned, they are not 'high' energy dogs, so do not require tons of exercise (ie labs, herding dogs). They have been bred to run for short bursts then rest for long periods. They do well in families, or with seniors and can adjust to many different lifestyles. Typically, they love kids and have been well socialized with other dogs (these can sometimes be problems with other breeds/shelter dogs).

Many retired racing Greyhound adoption groups exist around the country, so you can do an internet search for a local group, but here's a link to a few that will get you started with some information about the breed.

http://www.adopt-a-greyhound.org/

Go and meet the dogs, spend time with them, walk them on leash if that is allowed - at shelters, at kennels, at foster homes or wherever they are - before you make your selection. Each individual dog will have his/her own personality and although the breed and size do matter, the individual dog and the match with you and your family is the most important thing - ask the people who are caring for the dog, the shelter staff or foster family as much as you can to learn about your future pet.

Best of luck.

While it may seem that some breeds may bark more than others, it is really a matter of individual. I would spend a lot of time watching the dogs at the shelter before you adopt them. Look for ones that are quiet. When I adopted my two, I looked for one submissive and one dominant. They were both very quiet. Even the shelter staff commented at how quiet they were when I brought them into be spayed. There were rows of dogs barking and my two quietly sitting there.

Another thing you should consider is why dogs bark. If you give your dog enough stimulus they will be too content to bark.

Remember, how much a dog barks/acts out is more dependent on you taking the role of pack leader than it is on the dog's breeding. Remaining calm and assertive at all times is the key to that.

Good medium size dog would be a border collie. They are smart and respond well to training and have a high activity level. The longer the walk, the better. The best would be a mix with a border collie from a shelter. They run around 40-50 lbs. They are nippy so they are not good kids, they tend to want to herd them by nipping at their feet and hands. They may not be good with cats either. But if you want a tried and true breed, golden retriever is it. Ask around for a runt or small size 'cause they get between 60 and 100lbs. I find that females are the best.
Lastly, ask the girls and boys that work at the shelter where you adopt. They will tell you the different personalities of the animals there. It is always good to know what you are getting before you bring it home so visit the shelter for a couple of hours and take the animals for a "test drive" to see if your personalities click. Good luck! OH< also> contact your shelter and ask them who rescues animals in your area, they can put you in contact with some really helpful people who have the skills to help you find the right dog for you! 9 times out of 10 people surrender animals because they are not what they expected them to be. Be sure you know all the facts of the breed before you buy or adopt. I tend to stay away from "high energy" smaller dogs. (border collies are high energy but as long as you throw the ball in the back yard for about 10 minutes when you get home, they are good...plus the walk...)