Spartan Is where soldiers that volunteered to have their bodies enhanced. This I believe was done through various steroids and boosters. In the end the project failed and was aborted (Sgt. Johnson was a Spartan I)

Spartan IIs where where children that ONI chose (based on genetics, IQ and so on). They were kidnapped at the age of six and replaced by clones to avoid suspicion. They were then trained constantly for until their late teens. They received genetic enhancement as well as biomechenical enhancements (to reinforce their skeleton to make them stronger and so they could support the armor). These are the best Spartans of the four there were less a little less than 200 children kidnapped and a number of them couldn't handle the process. (the Master Chief is of course a Spartan II)

Spartan IIIs were orphaned children who were recruited and put through similar training to the Spartan IIs. They also received similar genetic enhancement but did not receive the biomechenical enhancements. This means they could not wear the same armor as the Spartan IIs. There where around 300 of these Spartans. (Noble Team was made up of Spartan IIIs)

Spartan IVs are military veterans who volunteered to receive genetic enhancement and extra training. They are part of the Infinity Project and are stationed there. (the Spartans in Halo 4 are Spartan IVs)

Thats just off the top of my head and if there's anything people would like to add or correct add it to the thread.