Definition: something extremely good, bad, strange, big, etc.
So, if, after going to the movies, I tell you that the movie was a real doozy, I guess your answer will be "Was is it a good thing or a bad one?" Like the case of 'funny strange, or funny ha ha?'.
I searched "doozy" on online dictionaries and, frankly speaking, did not quite catch the meaning.
Webster's dictionary online defines "doozy" as "extraordinary one in its kind".
Oxford Dictionary Online defines " doozy" as "something outstanding or unique of its kind" , which is  informal, chiefly  North American.
I suppose a broader context might help to understand its use better.
 Does doozy  have high frequency in English?
I've almost always heard this word in a negative sense to be something unexpectedly overwhelming. Â
"That first step is a doozy"
"That math problem was a real doozy"
"Her speech was not simple, it was a real doozy"
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asusena: I have heard this word more on TV comedies than in real life. Â
I believe "extraordinary" is the best one-word definition that has already been mentioned. Â It can be good or bad so context and implication are very important when this word is used. Â If you say a movie was a "real doozy" then more description would be required unless you are talking to a person who has also seen the movie, then the meaning can be inferred. Â My observation would say that its use is declining. Â Fifty years ago it was very commonly used in many situations but now it is probably true that it is heard more often on TV comedies. Â It hasn't totally died, though. Â In real life it is probably most frequently heard from people over 40 or 50.