RAID (redundant array of independent disks) is a storage technology that combines multiple disk drive components into a single logical unit so it behaves as one drive when connected to any other hardware. RAID 1 offers redundancy through mirroring, i.e., data is written identically to two drives. RAID 0 offers no redundancy and instead uses striping, i.e., data is split across all the drives. This means RAID 0 offers no fault tolerance; if any of the constituent drives fails, the RAID unit fails.
edit Data Organization in RAID 0 and RAID 1
RAID 0 offers striping with no parity or mirroring. Striping means data is "split" evenly across two or more disks. For example, in a two-disk RAID 0 set up, the first, third, fifth (and so on) blocks of data would be written to the first hard disk and the second, fourth, sixth (and so on) blocks would be written to the second hard disk. A downside of this approach is that if even one of the disks crashes, the entire RAID 0 setup fails because data becomes unrecoverable. In technical terms, this is described as a lack of fault tolerance.
A RAID 1 setup is different. There is no striping; the entire data is mirrored on each disk. This results in multiple copies of data (redundancy). And if one of the disk fails, data can still be recovered because it is intact on the second disk (most RAID 1 setups use only 2 disks, though some may use more), which means RAID 1 is fault tolerant.
Here is a good video that explains the difference between RAID 0 and RAID 1 arrays (a shorter video by the same person is on YouTube here):
RAID 0 offers very fast write times because the data is split and written to several disks in parallel. Writes to a RAID 1 unit is slower compared with RAID 0, but about the same as writing to a single disk. This is because the entire data is written to two disks, but in parallel.
Reads are also very fast in RAID 0. In ideal scenarios, the transfer speed of the array is the transfer speed of all the disks added together, and limited only by the speed of the RAID controller. Reads from RAID 1 may or may not offer such performance boost, depending upon the RAID controller. "Smart" controllers split the reading task in a way that takes advantage of data redundancy and reads different blocks from different disks. This offers a performance boost similar to RAID 0 but for controllers that are not capable of such multiplexing, read speeds and are about the same as a single hard drive.
edit Storage capacity
The total storage available for the RAID 0 unit is simply the sum of the storage capacities of individual disks because there is no redundancy. In case of a RAID 1 array, however, there is replication of data, which means the total storage capacity of the unit is the same as that of one hard disk.
RAID 1 is a better choice if reliability is a concern and you want to avoid data loss. A typical example is data archival needs. RAID 0 is a better choice in scenarios where a large volume of high-speed storage is needed. For example, capturing uncompressed HD video over HDSDI and recording it straight to a hard drive requires very fast writes and a large capacity. Another example is large databases that contain logs or other information that has a high volume of read operations.