RAID means "Redundant Array of Independent/Inexpensive Disks" and is mostly thought something you would do on hardware level via a separate RAID-controller or motherboard that supports RAID. RAID is used to get performance improvement for your disk operations, which is the biggest bottleneck in modern systems most of the time, or to provide redundancy when you lose your drive to hardware failure.
With Linux you can use something called 'software RAID' which does not require expensive RAID-controller nor support from motherboard. All you basically need is more than one hard drive. With two hard drives you can already setup a RAID0 or RAID1.
RAID0 means that you distribute your data on two (or more) drives and get a huge performance benefit, but the downside is that if you lose either disk to HW failure, you lose all the data on both disks.
RAID1 is quite the opposite of RAID0, RAID1 mirrors the data on first drive to second drive, which means that write operations are somewhat slower but the data is now duplicated and thus safe from HW failure.
There are several other RAID levels too, like RAID5 which requires 3 drives minimum and provides some redundancy in case of disk breakage and some performance improvements. My personal favorite currently is RAID10, which is a combination of RAID0 and RAID1, meaning that you get quite good redundancy in case of failures and quite nice performance boost also. The downside of RAID10 is that you need 4 drives to get started and 'lose' two of them for the mirroring.
SMART (or actually S.M.A.R.T) on the other hand means "Self-Monitoring, Analysis, and Reporting Technology", which is a technology most, if not all, modern hard drives support. With SMART enabled drives you can gather information straight from the independent drive(s) and use that information to predict when your disk fails of old age or otherwise. Hard drives have gotten lot better since infamous times of IBM 'Deathstar' hard drives, but still the fact remains: hard drives will die of old age sooner or later. If you can get a advance warning of this impending doom for your drive, you have time to make the necessary preparations, like making backups or even replacing it before you run into risk of losing your data.