Who came up with the handshake? And why? Why do some people bow to each other rather than shake hands?

The hand shake originally was a display that the parties weren't holding a weapon and were meeting in peace.

The other custom of bowing to each other was really each one inspecting the others shoe shine.

cooties

Both greetings probably came about for safety but for different reasons, the handshake to demonstrate you don't have a handy weapon, the bow to demonstrate trust because you're vulnerable. Also both can have secret meanings in how they are done. Masons, for example, had a secret handshake to identify what qualifications on a new job. It was resume of sorts.

No one really knows how it started, however the most common theory is that by putting your 'sword holding hand' out to someone shows that you are not a threat...and then they do the same back to you.

Bowing is traditionally more of a Middle East/Asian custom (although later taken over by other regions). Again, no one really knows the real reason why, but bending and exposing the back of your neck puts you at a significant disadvantage so you are acknowledging the power (and mercy) of the person you are bowing to.

I heard it was to show there was no weapon in the hand.

Not sure why came up with it, but they bow either because of that countries tradition or because hands contain a ton of germs