Some good info in this thread and some info that isn't quite right.
First off to the original poster . . . the quick answer as to what type to buy for your home. No question: ABC fire extinguisher. This is good for just about every type of fire you could expect to see in your home, it's easy to use and you can buy this type everywhere. As to what size . . . in general the bigger the better . . . the larger it is the more fire you can put out.
However, bear in mind that if you are using a fire extinguisher on a fire you should have a) called 911 to get professional help on the way (even if you think you can handle the fire), b) the fire should be smaller than you in size (if it's larger than you in general you should escape since the extinguisher may already be overmatched) and c) everyone in the area not using the extinguisher should be alerted and evacuating the area and d) do not use the extinguisher if you cannot stay below the layer of smoke.
There are many different types of extinguishers -- carbon dioxide, water, halon-substitute extinguishers, etc. but for the average home owner an ABC dry chemical extinguisher is the best all-around extinguisher to have around -- think of it as a Toyota Camry, Ford Taurus or Honda Accord in the fire extinguisher world. The other types are more or less specialized to certain needs -- think of them as the Volvo Box truck, Dodge Ram and Porsche Boxter (and not in any particular order) -- great vehicles/great extinguishers, but they're really made for specialized needs rather than just running the kids to soccer practice, picking up groceries, etc.
About the only drawback to the ABC dry chemical extinguisher is that it makes a wicked mess . . . it is made up of a very fine powder which can be hard to clean up (hint: use a shop vac with a dry wall filter . . . sweeping it up or vacuuming it up with a normal vacuum will result in powder flying everywhere.)
Contrary to belief, the chemical agent is not toxic or hazardous . . . but some chemicals have corrosive properties. Now I'm no expert, but if something can be corrosive I wouldn't advocate standing around inhaling great quantities of the stuff if you know what I mean. Folks who have respiratory problems, such as asthma, should not breathe in this stuff -- while not toxic, it isn't the best stuff to breathe in either and as a rule I don't like breathing in the stuff as it has an acrid taste/smell (trust me, I've breathed in a lot of this stuff over the many years teaching fire extinguisher classes.)
Incidentally, how the dry chemical works to extinguish a fire is a bit different than other extinguishers . . . the chemical basically breaks apart the chemical process which sustains the burning process vs. water which cools the ignition temp of the fire and carbon dioxide which suffocates the fire.
Oh yeah . . . as you no doubt know . . . the best thing for everyone involved is to not even have the fire in the first place by maintaining the stove (i.e. cleaning the chimney, proper disposal of ashes, etc.), installing the stove/chimney correctly (i.e. clearances, floor protection, etc.), running it properly (i.e. not over-firing it, forgetting to cut back the air, etc.) and knowing how your stove works and runs (i.e. knowing when and when not to add wood, cut back the air, etc.)