What do terms half-rate and quarter-rate mean for DRAM memory controllers?
From Altera's Using High-Performance DDR, DDR2, and DDR3 SDRAM With SOPC Builder:

Full- or half-rate SDRAM high-performance controllers have the
following definitions:

Full-rate controllers present data to the local interface at twice the width of the actual SDRAM interface at the full SDRAM clock rate.
Half-rate controllers present data to the local interface at four times the width of the actual SDRAM interface at half the SDRAM clock
rate.

Implementing the SDRAM high-performance controllers in half-rate mode
gives the highest possible SDRAM clock frequency while allowing the
more complex core logic to operate at half this frequency.

Basically, half-rate (and by extension, quarter-rate) application interfaces allow the SDRAM clock to be faster relative to your internal system clock. This improves both bandwidth and latency.