Nikon D5000's spec says this camera does not support SDXC, only SHDC.
I am confused about this card. Is it SDHC or SDXC?
"SanDisk Extreme SDHC 16GB SDSDX-016G" (30 MB/s speed card)
Is this card compatible with Nikon D5000?
It is compatible. All cameras supporting SDXC also accept SDHC and SD. SD cards are limited to 2 GB. SDHC to 32 GB and SDXC have a theoretical limit of 2 TB. SDXC memory cards also usually use a different file-system which is exFAT, rather than FAT.
In the case of 16GB, it is always SDHC. Although it would be possible to make a 16GB SDXC cards, no one does it and there is no point in doing so.
The converse is not true, so an SDHC camera cannot accept SDXC. Some manufacturers enabled compatibility via a firmware update.