We should make a disclaimer here that there's probably no such thing as the "wrong" bike for adventure cycling. The focus should be on the experience, not necessarily your equipment. If you wanted to unicycle across Uganda, for instance (not that we recommend it), you would probably garner a lot of attention and have lots to write home about. It just wouldn't be the most comfortable or efficient way to travel.
The point: try to suit the bike to where you'll be going. Mountain bikes, road bikes and tourers all have their strengths and weaknesses, which is why many dedicated cyclists own more than one type of bike. If you'll be sticking almost exclusively to paved roads, you can get away with riding a bike with skinnier, smoother tires that allow you to ride faster.
If your adventure is likely to traverse paths with dirt, gravel and possibly mud, you'll want a bike with better off-road capability -- starting with thicker and grippier tires. A full-blown mountain bike is probably too extreme -- its high-rolling resistance wheels would be a hindrance on the open road. But you can convert one to be more multi-mission capable by swapping out parts. Alternatively, you could buy a hybrid bike, which blends the strengths of road and trail bikes. While it excels at neither task, it doesn't suck at them either, making it a good all-around transport. But you might need to buy stronger rims to bear the weight of your gear.
Keep in mind the bike will be supporting more weight on an adventure cycling trip than it would simply ferrying you about town. A certain class of bikes -- touring bikes -- are purpose-built with this in mind. They boast beefier construction, built-in attachment points for carrying stuff, and extra clearance for your cargo. They also carry a beefier-than-normal price tag: $600 to $1,500 and beyond! For that reason, many people start with a more affordable bike and adapt it to touring by adding stronger rims, more versatile tires, and appropriate accessories.
Specific things to look for include a sturdy, steel frame (heavier, but easier and cheaper to fix than aluminum or titanium); front suspension to absorb bumps (but skip the rear suspension); and 26" wheels (easy to replace worldwide) if you plan to ride outside of North America.
Expect to pay around $200 or $250 minimum for a decent starter bike -- retail. Try to stick with well-regarded brands such as Trek, Giant, Cannondale, and the like, or specialty manufacturers that have a good reputation among cyclists. While these bikes are more expensive than your typical discount store offering, they more than make up for it in the quality of their construction. If money is an object and an obstacle, look on Craigslist or in your local classifieds before you plunk down your hard-earned coin at a bike shop. You'll be shocked and amazed by how desperate people are to part with name-brand, next-to-new bikes in exchange for very little cash. That little factoid, by the way, applies to just about all the equipment mentioned in this article!