What does it mean when a guitar pickup is deemed "hot"?
It tends to mean it is a higher output pick-up. This would drive your amp/gear "harder" than the non-hot pickup, giving a more dirty/distorted (some would say more modern) tone.
Like many things though, it is all relative. My Telecaster has "Hot" pick-ups, so they are more Rock than a traditional Telecaster pick-up, but they still have a lower output than a general purpose Rock pick-up like a typical EMG.
A hot output (as well as pickups, can be bass drum microphones amongst others) are outputs that are particularly loud, or produce much higher output than other signals. A hot pickup with overdrive applied to it would produce a much more distorted sound than, say, a regular pickup.
As runrunraygun has already said, some people call this 'Modern' or 'Rock' tone.
From my experience a pickup described as "hot" usually has more turns of wire on the pickup than usual--thus enabling a "louder" or "larger" amplitude to be induced on the pickup itself. This is a little subjective though, because one person can consider a Seymour Duncan SH-JB1 to be hot and another can consider a 5% overwound Lollar imperial to be hot--but they may not be equivalent in terms of impedance and signal production. A hot pickup usually overdrives your amplifier more easily due to the larger signal that is being driven to the amp.
An example is this: Humbucker pickups are generally way "hotter" than single coils obviously due to the construction of the pickup. A humbucker may have twice or more the amount of coil on the pickup itself because it's two single coils in series. This phenomenon is why single coils don't drive your amp to distortion as quickly as a humbucker will--and thus they usually sound more clear. However, you can overwind single coils to increase the "hotness" of them so they produce distortion more quickly.
The general rule I think is this: the larger the coil the hotter the pickup.
Hot is a relative term that refers to the signal output level compared to a traditional vintage pickup. Originally pickups were designed to produce a clean, balanced sound to faithfully reproduce the guitar. Higher output pickups (hot) are able to overdrive an amplifier more easily. Adding more winds to the coil increases output and resistance in general. They are not necessarily "dirty" unless the increased output pushes the amp into overdrive.