An alcoholic drink that has had something nasty added to it, usually some drug that will knock some one out.
Originally a Mickey Finn was specifically a drink that had been doctored with a strong laxative, which might be used by bartenders to chase off unsavory customers. We don't know who or what exactly the original Mickey Finn was, but most stories suggest that it was a bar or person from Chicago. It was first used in the early 1900s, and was quite common by the 1920s.
Nowadays, a Mickey Finn, or mickey, most likely refers to a drink with chloral hydrate added, although it could be used for any drink with drugs to disable the drinker.
On-line slang dictionaries (not always reliable, but always fun) inform me that it can also be used to refer to depressants of all sorts, or used as a verb meaning to leave ('to take a Mickey Finn').