Brann is unique in that almost all that we know about him has been told to us by him. Moreover, most of what we know about the furthest reaches of Azeroth has been told to us by Brann via his writings.
Brann's 'early' years, the first major chapter of his life that we know in detail, occurred during the Second War. With his brother Muradin, he served the Alliance forces alongside Anduin Lothar and his second in command, Turalyon. The two would serve as official liaisons between Khaz Modan and their allies.
Around the time the Third War came to its conclusion, King Magni sent his little brother out into the world to fulfill the purpose of the Explorer's League: Explore, discover, and map out the furthest reaches of Azeroth, and begin searching for the origin of the Dwarven people. Brann's journey would begin in the Eastern Kingdoms, the land most familiar to him and Dwarvenkind, and would then move out to progressively more 'unknown' regions. To Kalimdor, then the Isles of the South Seas, and finally up to Northrend.
Brann Bronzebeard has this habit of going missing, you see. He would never admit to being lost, and maybe he never really was lost, but he was pretty good at ending up 'missing'. His career of exploration began this way, and it has continued ever since. His career began in the relatively tame regions of Elwynn Forest and Westfall, though tame is relative in Azeroth.
His first 'whoops he's missing' moment came shortly after entering the jungles of Stranglethorn. His communications with the Explorer's League stopped shortly after coming upon the remnants of Zul'Gurub, the capital of what remains of the once-great Gurubashi Empire. What happens to him here is largely unknown, but eventually he shows up alive and well somewhere completely different. His travels also carried him through Khaz Modan and into Lordaeron, and the remains of Quel'Thalas.
Having concluded his adventures in the Eastern Kingdoms, Brann resolved to head over to Kalimdor. Kalimdor's existence had only recently become common knowledge to the people of the Alliance, so Brann's adventures were far more varied and interesting. New races to discover, new languages to learn and dissect, and all sorts of tasty potential digsites for Titan artifacts.
'Whoops he's missing' event number two came around when Brann was en route to Northrend, but like before, he eventually cropped up again. Brann originally traveled to Northrend to find out what really happened to Muradin. You see, shortly before Prince Arthas Menethil returned to slaughter the people of Lordaeron, King Magni received a polite little letter.
"To King Magni Bronzebeard, Lord of Ironforge,
It is my sad duty to bring you very grim news. I am certain you know of Muradin's mission here in Northrend to strike at the Scourge and discover useful artifacts. We fought together against the Scourge, and although I survived, Muradin fell to the undead and demons.
I grieve for my brave friend and your valiant brother. Yet know that his heroic death was not in vain, for his life ensured my victory against the Lich King's minions - and my recovery of the ancient sword known as Frostmourne.
I shall be returning to Lordaeron soon. With Frostmourne in hand, I come to reestablish order and bring forth a shining new age. Muradin's remaining clansmen here will take his body back to Ironforge.
You cannot know how deep are my sympathies. I know that this letter offers small consolation, but I felt you should learn of Muradin's death as soon as possible. You have lost a brother, and I have lost a valued friend.
May the Light preserve our peoples,
Prince Arthas Menethil"
If you are really behind the times, I'll help you catch up a little: Arthas gets Muradin killed, Arthas kills King Terenas Menethil II, Arthas ruins Lordaeron.
I think you can guess that this letter is a little suspicious, no?
While in Northrend, Brann made a number of discoveries, and forged many bonds. Not only does Northrend contain many promising leads for the possible Dwarven link to the Titans, he also met many very unusual races. The Tuskarr, the Magnataur, and of course, he weaseled his way into the Sundered Monolith, the last bastion of Nerubian civilization after their defeat at the hands of the Scourge.
After Brann's first round of travel, his break was brief. He entertained the notion of settling down and teaching history to the younger Dwarves of Ironforge, but his travels began anew. During a more thorough exploration of Kalimdor, Brann found himself a partner in the form of a monkey named Glibb. Dwarves are smelly, monkeys are smelly, they were a perfect pair!
Brann became fascinated with the Qiraji and the Scarab Wall, likely driven by his encounter with the Nerubians and his obsession with history, and worked diligently to find a way into Ahn'Qiraj. Then he went missing for a third time. He found a way in, it seems, and accidentally fed a very important artifact to an Old God. Whoops! Teehee~!
Thanks to Brann Bronzebeard, the base of Medivh's staff, Atiesh, found its way into the belly of C'Thun. Well done, my Dwarven friend. Well done.
Once the Scarab Wall of Ahn'Qiraj had been opened properly by an army of adventurers, Brann was able to escape, entertaining thoughts of finding a way to travel to Outland. We don't know whether he truly made it there or not, but as far as I can recall, we've heard nothing from him in our time there.
It seems that in Wrath, we'll encounter the eccentric explorer during his second trip to Northrend, doing who knows what. It's very likely Brann's presence will have something to do with the Titans and the Iron Dwarves, but he may also be doing something silly like trying to figure out how penguins slide so well on their bellies, or teaching giants how to love.
Only time will tell. For now, let's just hope he hasn't lost an essential artifact in the belly of the Lich King, too.
If you want more information on Brann Bronzebeard and his travels, I strongly recommend picking up any of the Warcraft RPG books that have been released since Warcraft III. The RPG itself often gets mixed reactions, but most of the information on the world is given from the point of view of Brann Bronzebeard, with his own opinions and commentary thrown in. Definitely check them out. You'll definitely learn about more than just Brann, you'll learn about Azeroth as a whole.