SD, SDHC and SDXC are all variants of the SD (Secure Digital) format, that have been released in that order by the SD Association.
As requirements for higher capacities came in, SD cards have grown in capacity too (SDHC is SD High Capacity, while SDXC is SD Xtended Capacity). SDHC cards typically come in capacities between 4GB to 32GB. SDXC cards come in sizes of 32GB of higher, with a theoretical maximum of 2TB.
Backwards compatibility is built in, so if a camera has SDHC support , it will support standard SD cards too. If a camera or device has SDXC support, it will support SD and SDHC. These cards have speed ratings too; Class 2, 4, 6, 10 and so on, with Class 10 having the fastest read-write performance.
MMC or MultiMedia Cards may look similar to SD cards, but they are an older format. If you look carefully, you'll notice that MMC cards do not have the tiny lock slider switch that SD cards have; the lock prevents data being written to or deleted from the card.
Most cameras that support SD cards will also support the use of MMC cards.