Finding the perfect breed?

Check out greyhounds or whippets. Although they are big dogs, great danes would be great too.

My first thought was "greyhound" but I don't know how well adapted they are to apartment living, there are loads of others on here who know more about this breed than me so I am sure they will be along to tell you more....  They don't need a lot of exercise but they do love to run and I think one would need a safe enclosed garden to allow that.  They certainly are not small dogs, but in my experience they take up less room than small dogs!  Because they are quite happy to snooze for a lot of the day, provided they get a walk morning and evening whereas a lot of smaller dogs are much more "on the go" a lot more of the time.  You just need to have a corner of the sofa reserved for them!
 
On the up side, they are generally easy to house train, docile, calm, and are not (in my experience) dominating at all.  I think they are only hard to train if you use the wrong approach - I think they respond best to motivational training, you have to be interesting and find out what they will work for, what makes them tick.  I think they get on OK with strangers but can come across as a bit aloof with people they don't know.  They are dead affectionate (again only my experience)  but are not as demonstrative as, say, labs or retrievers.  They  and are much quieter in their expression of affection - they like to cuddle up against you and snuggle!
 
They are fairly low maintenance, usually quite healthy, don't need much exercise or grooming, they are gentle, lazy and loving.
 
Downsides would be: never to be trusted 100% with recall and let loose in open spaces; they have quite thin skin and thin fur so be careful where you let them run so they don't injure themselves; they are not outdoor dogs and I shouldn't think one would EVER adapt to outdoor living (although that doesn't sound likely in your home); they NEED to share your sofa!
 
I will post back if I think of any other breeds, but in the meantime I would consider going to your local shelter and acquiring a cross breed.

I would say perhaps some sort of small terrier or terrier cross...maybe a mini schnauzer. While they are not known for being easy to train, terriers are incredibly smart and can generally be taught all sorts of things if you have patience. My mini-schnau lives in a 2BR condo with me, my fiance, 2 cats, a malinois, and numerous foster cats, making it a bit cramped, but she does great. She does go for a walk once a day (to exercise) and goes out numerous other times for potty breaks, but all in all, no shedding and she's very affectionate, adaptable, and a joy to have around.

Downside: As a pup, she was more difficult to housetrain, thanks to a bladder that is the size of a pea. Terriers can also be barkers if left to their own devices, but I've heard cairns and westies make decent apt. dogs if given sufficient outlet for their terrier-ness. []

I second the suggestion to go to your local shelter. Tell them what you are looking for and I bet they can find something that is up your alley. []

A Boston Terrier would be good easy to train very friendly, mini dachshund would be good also although not easy to house break.

I"m not sure what easy to train means. If you don't plan on teaching your dog circus tricks and have no aspirations to run the next great agility dog, most any dog can learn the basics fairly quickly - don't jump on people, sit when asked, walk on a leash nicely, generally come when called.

Unless you mean you don't intend to spend much time in actual training at all and hope there's a breed that sort of trains itself?

To me, someone living in an apartment has an ideal situation for dogs that can't ever be offleash. You'll need to dedicate more time than just one or two walks a day - or go farther than around the complex or up the street - but dogs like hounds and northern breeds love walking with their folks and wouldn't be safe to amuse themselves in a backyard no matter how well-fenced.

Jack Russell and other working terriers are NOT apartment dogs, except under similiar circumstances as above. Like living with Border collies or Aussies in an apartment, it takes a special kind of person who plans to center his free time on his dog, basically.

I think this really is a situation where you are going to have to go when you are ready and just check out what strikes your fancy at the local shelters and rescues. Don't forget to check out petfinder.org to make sure you are aware of all the rescue groups in your area. They don't all advertise. You can plug in your zip code and get a list of many, many dogs that need homes in your area. If you are gravitating toward a certain breed, you can search by that, too.

I guarantee you'll be shocked at how many dogs need homes that fit your particular circumstances - and I hope you can give one a home.

To me, someone living in an apartment has an ideal situation for dogs that can't ever be offleash.

OMG Becca we had a long-distance mind-meld! I was just saying this to someone the other day! I was talking about sight and scent hounds but it applies to northern breeds too: They tend to make good way out in the middle of nowhere country dogs OR great city dogs, but not so good in the suburbs because of the leash situation. Suburban dogs tend to be expected to be able to sit in a manicured back yard (sometimes not even fenced!) for hours at a time and not roam. Many breeds of dog just can not handle this. But when you live in an apartment or just a really dense section of city, it becomes a non-issue that you've got a breed that can't be off-leash. Everyone's dogs are on leash (or should be) and you don't even notice that there's a difference between the lab next door and your greyhound in that regard.