This is a bit of a challenge to describe but I shall try.
A single phase and a 3 phase alternator are vastly different in the way the internal poles are arranged and the number of coils and wires leading to the outside world. Basically a single phase unit will have two output wires and for every revolution of the shaft you will see either a single 360 degree sine wave come out (2 pole 3600 RPM head) or two 360 degree sine waves in the case of the more common 4-pole 1800 RPM heads discussed here. The single phase output winding is often center tapped to make 120-0-120 volts available from three wires.
The 3-phase head has three output windings instead of just one of the single phase head. Each winding of the 3-phase head will also make either a single 360 sine from each 'pair' of output terminals in a 2 pole 3600 RPM head, or two sines waves per rev in a 4 pole, 1800 RPM head, BUT the sine wave will be seen to begin on phase one winding, then 120 degrees of rotation later another sine wave will begin to be generated on the second set of wires and 120 degrees later yet a 3rd sine wave starts on the third winding. If you look at only one winding output it will resemble that of the single phase machine output exactly. In the three phase you have 3 simultaneously generated sets of sine waves that are each delayed or time shifted 120 degrees to the output seen on the other windings. The 3 separate coils create sinewave peaks can be traced back to coincide with 3 separate points, 120 degrees apart around the revolution of the armature in a 2 pole machine. This is repeated twice in a single revolution for a 4 pole machine where one 360 degree physical shaft rotation actually represents 720 electrical degrees.
Confused yet?
Maybe this will help to form the image. You can generate 3-phase power IF you have three identical single phase heads and arrange their shafts to be phase locked to each other through a gear system or perhaps toothed timing belt. The keyway on each shaft can be used as a physical phase reference. The keyway on the first would be set at 12:00 o'clock. The key of the second would be set at 120 degrees later or say 4:00 o'clock. The key of the third shaft would be locked 240 degrees later or at 8:00 o'clock. As the prime mover rotated these three heads they would always have the same 120 degree phase or timing separation between their armature shafts. The six total single phase output wires from all three heads would give you true 3-phase power. Wire them all in series would give you 3-phase Delta. Wiring them all with one lead common makes a neutral and three live wires. This is three phase Y or Wye. In 3-phase Wye, the common 4th wire is used or not depending on the application.