Why do we use 3-phase systems ? or what is the advantages of the 3-phase systems on the single phase systems ?

you may want to search Wikipedia for polyphase electrical systems.. there is probably a whole chapter in a power engineering textbook devoted to this topic.

3 phase can transfer more power with the same amount of wire copper used - more economical.

3 phase motor run smoothly, smaller size, less vibration. better power factor, etc

Two reasons:

1. Constant power. You can mathematically prove this (yikes!), or look at some proofs online if you'd like, but the big problem with single phase is that you get 120 instances per second with zero current flow. This stinks if you're powering something like a motor and don't want a jerking motion.

2. Bang for the buck. Just look at the basic math using single phase as the baseline:
Single phase: 2 wires, 100% power delivered
Three phase: 3 wires, 173% power delivered

That's 73% more power for 50% more material cost.

yeah 1.732 is the answer!