So many RAID levels, and mixes.
Intel matrix RAID, for instance, is pretty versatile in the kind of volumes it can handle and build usi9ng whole drives or parts of drives.
Just as with hardware versus software modems, it seems there are the same distinctions:
"Hardware RAID" would have the parity calculations in hardware.
"Software RAID" would do it all in software.
Motherboard RAID tends to be inbetween, if using for RAID 5, it would be calling on the host CPU for the parity calculations.
Now RAID 0 and 1 do not use parity, RAID 0 is disk striping, RAID 1 is mirroring, and the levels can be combined - with 4 drives:
Raid 0+1 (mirror of stripes), or RAID 1+0 (stripe of mirrors).
Pretty sure most will also allow you to make two seperate arrays - one use being to make a fast RAID 0 stripe for OS and apps, with a RAID 1 mirror set for data.
The onboard controllers tend to have a bus efficiency advantage due to their closer coupling with the chipset, so will often be better than anything that isn't a "smart" (and expensive!) add-on card.