I have a 2005 Ford Expedition XLT with 145,000 miles on it. I have water coming into the passenger side cabin from a 'black box' (see pictures). The box has a crack in it. My question is what is this box called? And if I replace/seal it, will the water issue be resolved or must I further investigate if the port that this water is to be draining to is clogged? Thanks you for any suggestions.

 

That box is your evap case, there should not be a crack in it, keep in mind it is two pieces which screw together so there is a seam though. After fixing the water issue you could use a body seam sealer adhesive to fix the crack.
When the AC is on condensation forms on the evaporator, depending on the humidity this can be a lot of buildup which must drain from the vehicle. If you park your vehicle on a dry flat area and let the ac run for about 5 minutes you should see a puddle of water forming under the vehicle, if there is no water dripping from the vehicle then your drain tube is clogged up, you will need to access it and clear the obstruction but be ready for a rush of water once its cleared, I like to use a blast of compressed air to knock the obstruction loose and let it flush out with the water that comes out. I dont know where the tube is on your vehicle, they are often behind the center console side panel and route to above the heatshield under the vehicle.