At their most basic, RAID works in principal the same way whether it is hardware based or software based. The main difference being all the processing required for compression, parity calculation (encryption if any), drive cueing and such are handled in silicon, offloading those chores from the CPU and providing a noticeable speed improvement at the same time.
"Fake-RAID" is a combination of on chip BIOS instructions and OS drivers that do fundamentally the same thing as a hardware RAID controller but far more cheaply and somewhat slower.
Software RAID does all the operations in software using the CPU to do any needed calculations. Again it does the same fundamentals but more slowly.
The tradeoffs are simple:
1) hardware RAID is fast but costly. Drives configured using one RAID controller generally only function as intended when connected to a controlelr of the same type.
2) fake-RAID is moderately speedy and fairly inexpensive but is more resource hungry and usually quite proprietary to the motherboard or controller card used. Moving disks from one fake-RAID system to another often does not work.
3) software RAID is the slowest option but has the benefot of being very transportable between machines so long as the same OS is in use.
RAID 1 is the simplest form of RAID (disk mirroring). Generally all that you need to do to restore an array (so long as you have one good disk) is to remove the defunct drive, add a new one of equal or greater size (formatting if necessary to the RAID system you're using), add the drive to the array and the RAID hardware and/or software should rebuild the array by creating a copy of the functioning drive to the new one.
If both drives have failed of course the array is irretrievable and must be recreated from scratch.
There are many descritpions on-lien of the various forms of RAID and how they work. A decent set of visual tutorials on the most common RAID levels can be found at
JetStor - Visualize Your RAID
Hope it helps some.