Error message: qgis_core.dll missing

I also receive the same when opening .shp files directly and selecting qgis-bin as the program to use. But I always load QGIS via the Desktop icon. I'm not exactly sure why but my guess is that the Desktop icon simultaneously runs the following 2 files in order to load QGIS (this is the Target which you can see in the icon's properties):

"C:\Program Files\QGIS Valmiera\bin\nircmd.exe" exec hide C:\PROGRA~1\QGISVA~1\bin\qgis.bat

My advice is to load QGIS via the Desktop icon (usually has a name like "QGIS Desktop 2.4.0") and then drag/drop your shapefile that way.

Edit:

As Steve has already found out, it's possible to drag your shapefile to the "qgis.bat" file which will load QGIS and the shapefile itself.

I had this issue recently. None of the answers here worked to me.

When I use QGIS 2.4, I usually work with my laptop plugged in a bigger lcd monitor. Then, I project everything in the second monitor using a hdmi adapater. So, when I turned off the second monitor, QGIS started to show the message 'qgis_core.dll missing'.

To fix, I pressed the key 'Fn + F1' (Dell laptop) and choose the first option to project images only to the first monitor. Than, the QGIS worked again.