You know, of course, that dwarves are just about 3 and a half or 4 feet tall, with craggy skin you could probably sharpen a knife on. Dwarves are also able to periodically harden their skin to be much more like stones than normal. Most dwarves you see tend to have lots of hair, with not a male in sight who doesn't take pride in his beard. Even some female dwarves are said to have beards, though we haven't yet actually seen any bearded ladies in the game. Their skin color ranges from pale tan to dark brown just like humans, but also includes all the shades of grey from chalky white to charcoal black.
Dwarves are the first race introduced in this series so far to have a lifespan quite different from humans, which causes certain problems as far as roleplayers are concerned. First of all, we know that dwarves reach "adulthood" at about 40 years of age, reach "middle age" at about 125, and finally get "old" at about 188. A venerably ancient person might live as long as 250 years. This seems seems pretty simple on the surface, but it raises questions like, "is this 'adulthood' actual physical maturity, or is it merely the age at which dwarven society considers people adults in a social sense?" The way the ages are given in the Warcraft Pen and Paper RPG book seems to indicate the former, that dwarves the age of 25 or 30 might be going through puberty of a sort. What puberty might feel like after going through 25 or 30 years (rather than just 10 or 12 for us) is up to you to imagine. (It might not seem too hard to imagine now, but wait till we get to extremely long-lived races later on.) In any case, if you've read The Hobbit, by J.R.R. Tolkien, then you probably remember Bilbo Baggins feeling that 50 years of age is not so old, certainly not too old to be going on adventures as a burglar. It seems life just moves at a slower pace for hobbits, and for dwarves in World of Warcraft as well, and as roleplayers it's important to pay attention to that. A dwarf ought not to go around saying "Aye lassie... Ye be lookin at a handsome 25 year old!" as this would be tantamount to, "Hey check me out ladies -- I'm 9 years old!" for a human.
Being a dwarf is all about your connection to the earth, though not in the same sense as some other races might think. The dwarven approach is inherently scientific. Although the dwarves have a strong connection to the Holy Light like humans do, their main passion for countless centuries has been mining and engineering. Discovering the hidden properties and uses of various metals and ores feels to them like discovering some part of their own nature. As dwarves discover, metals and ores are a part of their nature, and they take pride in this fact.
Your characters' parents or grandparents might have been around during the time when all three dwarf clans were still united and ruled by the great king of Ironforge, Modimus Anvilmar. When he died, however, civil war ensued -- the Bronzebeards, the Wildhammers, and the Dark Irons all fought for possession of Ironforge, with the Dark Iron dwarves revealing themselves as the nastiest of the lot. Now, although the ties between the Bronzebeard Clan (also known now as the Ironforge Clan) and the Wildhammer Clan have mostly smoothed over, the Dark Irons are still dire enemies of the other dwarves, and continue to practice great evil under the bowels of Blackrock Mountain, not too to the south of Ironforge Mountain itself.
But all that would have been the talk of previous generations. Growing up in Ironforge city, your character would have been immersed in your studies of all things earthy until the great events that rocked the whole world during the Second and Third Wars, when your long-time allies, the humans, called out for your aid, and you helped them drive off the orcs, demons and undead.
During those upheavals, which we have discussed previously, your character might have had the opportunity to travel farther than he or she could have ever imagined growing up. As the dwarven people opened their eyes to the world, they were amazed to discover hidden ruins and ancient relics which contained hints to the mystery of their creation.
Initial reports from the field of archeology, which your character would have undoubtedly heard, indicate that the dwarves originated as one of the first creations of the Titans who shaped Azeroth. Back then, they were called "Earthen" and were entirely made of stone, not flesh as they are today. What caused them to change into their current state remains a mystery.
Just a few years prior to the beginning of World of Warcraft, King Magni Bronzebeard heard about these great discoveries and declared a bold new direction for the dwarves -- possibly the greatest shift they've ever taken in their society -- to move from mining and engineering into archeology, and uncover everything they can about their origin and true nature. Of course this is not to say that they left mining and engineering behind completely, nor is archeology such a great departure from other forms of digging in the earth. The difference is that now dwarves have a kind of manifest destiny, a birthright which entitles them --indeed demands of them -- to explore, to seek out, and to discover the truth of what makes them who they are. This impulse has driven dwarven expeditions out of their secluded state and into the vastness of the world, as far afield as the Badlands, Tanaris, Mulgore, and even Northrend.
Every dwarf knows of the great Bronzebeard family: King Magni Bronzebeard is the great ruler of the dwarves; his younger brother Muradin Bronzebeard was the founder of the Explorer's League, a great hero in the Alliance, and is believed to have been slain in Northrend at the outset of the Third War; and finally Brann Bronzebeard, the most famous, most mysterious, and most widely-traveled explorer ever known in Azeroth.
You may also dream of being a great hero of the dwarves, like the Bronzebeards. Whatever professions your character chose (even if these are other than mining and engineering -- someone's gotta plant the flowers and stitch the clothes!) he or she likely wants to achieve great excellence in this great quest of the dwarven people around the world. They are excited to the bone to be part of such a grand enterprise, whether through diligent study, the vanquishing of great enemies, the exploration of unknown lands ... or even just the digging through masses and masses of mud.
For more reading about the mystery of the dwarves' origin, or their history, check out their page on WoWWiki, or even their page on Answers.com for (surprisingly) a bit more detail.