Disconnect all other external devices to see if one of them is causing a problem. It is known that a Wacom tablet can prevent the mounting of some external drives, but the same may also occur with other connected devices. I've seen reports that disconnecting the Wacom tablet will allow the drives to mount, then connecting the Wacom tablet to use both the devices.
Also, make sure the drives are connected directly to the Mac since some adapters, docks, & hubs may cause problems.
Try booting into Safe Mode to see if you encounter the same issues. Keep in mind Safe Mode disables third party drivers from launching so if your external drives are utilizing the manufacturer's proprietary software, the drives may not mount.
What is the exact model of your Mac? You can get this information by clicking the Apple menu and selecting "About This Mac". Also what are the make & models of the affected drives and the file systems on them? Do they have their own dedicated power supply or do they rely on bus power? This is all information that should be provided when asking for assistance.