Why does my solder occasionally pop?
As the other answerers have said, the clear, hard substance is the rosin from the solder.

There are a number of things which can cause what you describe.   


Voids in the rosin core:
 As the solder heats up, the gas in the void becomes pressurized. Once the solder is sufficiently molten that the gas in the void can escape, it does so. Violently.
Some rosin is hygroscopic:
The composition of some rosin cores can cause it to absorb moisture from the air. Once the rosin is heated to a certain point, it releases the absorbed moisture in the form of steam. Sometimes rather violently.
Hot-Spots on your soldering iron:
If you are using a low-quality soldering iron, the tip may not be very thermally conductive. Therefore, if you apply a lot of solder to the tip, it can cool fairly dramatically. If the applied solder then runs along the surface of the iron (due to surface tension) to a section which is much hotter, it can cause the rosin to boil off suddenly.


Overall, the little rosin balls are generally fairly harmless. If you want your boards to look really nice, you can wash your assembled PCB once you are done putting it together.
Look up your flux's datasheet. It will tell you what you should clean it with. There are water soluble fluxes, as well as more aggressive fluxes which require solvents.

From your post, ("44"), I assume you are using Kester #44 solder. You can find it's datasheet here.

It states:


  Cleaning:
  
  Kester 44 possesses excellent fluxing ability, the flux residue is non-corrosive and non-conductive under normal conditions of use.  When exposed to an elevated temperature and humidity environment (38°C, 94% RH) for 72 hours, there is no evidence of corrosion caused by the flux residue.  Throughout its many years of wide usage, 44 Rosin Flux has produced many billions of soldered connections.  In all these billions of solder joints, involving the most delicate and critical of electrical and electronic components, there has never been an authentic instance of corrosion by the flux residue under normal conditions of use.  This mild property of the residue permits leaving the flux on the assembly for many applications.




It's worth noting that more aggressive fluxes can be harmful to leave on your PCB. Such fluxes can actually eat away at the plating on the device pins over time. 

As a last note, one area where cleaning the flux off the PCB is critical is very high impedance applications.
While the flux residue is a good insulator, it is significantly worse then the PCB alone. Therefore, you can have considerable leakage through the flux (~1MΩ between SMD pins). While it is not much current, if you are trying to measure nanoamperes, it can be a significant issue.