How can I enable onboard graphics AND dedicated card simultaneously?

To flesh out Sathya's answer a bit: In most systems, the same PCIe lanes are used for both the IGP and the PCIe-x16 slot for the video card. So either the slot can be used or the IGP. This means you can't even put non-video card devices (e.g. RAID controllers) into the x16 slot without losing access to the IGP -- you'd still have to install a video card in a different slot!

Late Edit: It appears that on Sandy Bridge systems with an H67 (and probably Z68, when they launch) chipsets, it's possible to run both the onboard GPU and an add-on graphics card at the same time. Other 6-series (and later) chipsets may work as long as both the CPU & MB Chipset support Intel's Flexible Display Interface, a DisplayPort-based standard that gives the integrated GPU a direct connection to the onboard video connectors.

Unfortunately, there's no way around this limitation - using an external card automatically turns off the IGP.

In Asrock UEFI Bios of the h81m-vg4 motherboard there is an option to enable the onboard graphics while you have an additional video card connected:

Advanced > chipset configuration > IGPU Multi-Monitor > Enabled.

The h81 chipset allows the flexible display interface. I tested with the intel g1820 and the msi r5750 and I'm able to run up to 4 independent displays.