In HD2 there's no auto-downloading (installing) possible. So we need to do some handwork. But don't worry, it's simple.
Now basically you need to download a custom mission, unzip it (as in most cases the files been archived) and then copy the new files into your HD2 main directory.

HD2 main directory by default (64-bit Windows) is located here:
C:\\Program Files (x86)\Illusion Softworks\Hidden & Dangerous 2
and for 32-bit Windows:
C:\\Program Files\Illusion Softworks\Hidden & Dangerous 2

Detailed

If you have sucessfully unzipped the mission files you should have got following folders
  • Missions
  • Scripts
  • Models
  • Maps
  • Sounds
Now just move these folders into your HD2 main directory.
To check if you have all done correctly you're folder structure should look like below.

C:\\...\Hidden & Dangerous 2\Missions\MISSION_NAME (contains scene.4ds, actors.bin, scene2.bin, tree.klz, ...)
C:\\...\Hidden & Dangerous 2\Scripts\MISSION_NAME (contains .scr files)
C:\\...\Hidden & Dangerous 2\Models\MODEL_NAME.4DS
C:\\...\Hidden & Dangerous 2\Maps\IMAGE_NAME.BMP
C:\\...\Hidden & Dangerous 2\Sounds\SOUND.WAV

Additional files like .txt files as the ReadMe you don't need to copy. However if you want to play a custom mission
on your own server then you have to update your mpmaplist.txt file which is in your HD2 main directory.
Update maplist tutorial