In reference to memory cards, HC stands for High Capacity.
HC cards can have larger memory capacity than non-HC cards. Non-HC memory cards can only store up to 2 GB. HC cards can be any size from 4 GB to 32 GB.
HC involves specific technology that enables capacities larger than 2 GB. Devices that accept memory cards must be HC-compatible in order to read HC cards. Devices that are HC-compatible are also backward-compatible with 2 GB and smaller cards.
HC cards come in multiple sizes, including full-size SDHC and the very small microSDHC. The key difference is physical size. Cheap adapters are available that allow microSDHC cards to fit into SDHC slots.
A newer standard called SDXC supports even larger storage capacities, up to 2 TB.