The EPS12V processor power connector is intended to provide additional power specifically for use by the processor, as opposed to simply using the primary 20+4-pin ATX power connector. This connector can have 4 or 8 pins, with the first 4 pins of the 8-pin version being identical to the 4 pins in the 4-pin version. As such, power supply cables for EPS12V are often configured as a split 4+4-pin connector that can be used together or separately as required by the motherboard.
The more power the board is designed to handle, the more likely you'll need the 8-pin connector, and some high-end motherboards may even have multiple EPS12V connectors. My system has an 8-pin connector and an additional 4-pin connector, though the motherboard manual states that connecting only the 8-pin connector is mandatory; the extra 4-pin connector is optional and is there to help supply more power for overclocking.
4- and 8-pin EPS12V connectors on the ASUS ROG Crosshair VI Extreme motherboard. Notice how the 4-pin connector on the board is connected to one half of an 8-pin cable; the other half is left disconnected.
In your case, plug one half of the 4+4-pin EPS12V connector from the power supply into the motherboard (only one of the two halves will fit). The other half of the connector can simply be left disconnected.