RAID-0 gives you extra storage, but no backup ability, like two bottles stacked on top of each other.
RAID-1 mirrors the drives, so if you lose one drive, you have another.
RAID-5 uses three drives, a left drive, a right drive, and a parity drive. As long as you have two of the three drives, your data is safe. 'Tis why I use RAID-5
RAID 0+1 is simpler, it's doubling drives with parity... but unless your drive controller is really stupid, using RAID-5 gives you more space than mirroring.
Loved the picture, but I'm going to expand on this just a bit.
RAID-0: Fast, but no redundancy. Data is alternately written to drive 0 and then drive 1, meaning that writes and reads are less adversely affected by spindle speeds and seek times. No data protection at all. Best used for those who simply feel that they have to get the absolute best performance possible; those people who want to eek out that extra 5% of performance from their uber God box. If any one drive is lost, data integrity cannot be maintained.
RAID-1: Writes are technically slower, but not always if you have a good hardware controller card. RAID 1 is basic mirroring and the controller writes the exact same data to both halves of the mirror. Typically done in 2 drive setups, as a method of cheap, quick data redundancy. Cheapest method of data redundancy on a small scale. Larger scale always doesn't make sense as you "lose" half your raw disk space for redundancy. Can sustain one drive loss and still maintain data integrity.
RAID-5: This is the good stuff and probably the most used RAID type at the enterprise level as it is the most economical use of HD space. A basic RAID 5 setup consists of two data drives plus a parity drive. Data is written alternately to each data drive, while a parity bit is written to the parity drive for each write transaction. Technically write speeds are going to be slower since the parity drive is written to for EVERY write transaction, but with a good controller, there should be little loss. The biggest advantage of RAID 5 is that, depending on the size of your disks and number of your disks, you will "lose" a minimum of space for redundancy. For example, one of my RAID arrays at the office consists of 8 500 GB drives. In that array, I still only need one of those drives for parity information, so that means I still have 7 X 500 GB (3.5 TB) worth of space. I think you can see why it's more economical than the straight mirror, RAID 1. Can sustain one drive loss and maintain data integrity.
RAID-0+1: This... actually, I'm not sure why anybody uses this.

The speed advantages achieved due to the striping will be outweighed somewhat by the fact that you have a second striped set to write the exact same data to. Really, at this level, the performance gains you see will be heavily dependent on the controller, but there will be some advantage over RAID 5, as the controller won't have to calculate any parity information. This is most often used in situations where a particular controller can't do parity calculations, but when more space is still desired in the RAID array. For instance, my RAID card in my desktop isn't RAID 5 capable, so if I wanted to expand my storage space with another pair of drives, it would only make sense to get the added performance benefit from striping each half of the RAID 1 mirror. Can sustain one drive loss on each half of the mirror while maintaining data integrity.
There are other RAID types out there, but those are your four basic types of RAID that are most commonly used.