What is the significance of shouting "Hosanna" by the people of Jerusalem? The word Hosanna has many meanings: "help", "save, I pray", "Save, now!" or Praising etc.

If the people shouted "Hosanna" in order to praise Jesus merely as a great Prophet, it would be easier to shout "Crucify him!" later when they were disappointed by his acts.

If the intention of the people of Jerusalem was to say "Save us from the hands of Rome", it means they accepted Jesus as the Messiah after seeing all the miracles he performed. But in less than a week they became impatient and shouted "Crucify him!", thereby rushing to a conclusion that Jesus was actually not the Messiah. 

What was the intention of the people of Jerusalem when they shouted Hosanna?
Many of the people of Jerusalem indeed thought that Jesus was the Messiah. There were so many prophecies that were fulfilled by Jesus, and the common person at least had knowledge of some of these prophecies, as they were what many of them had to look forward to.
On Palm Sunday, Jesus fulfilled one more prophecy (Zechariah 9:9) that many would not have missed:

Rejoice greatly, O daughter of Zion!
Shout in triumph, O daughter of Jerusalem!
Behold, your king is coming to you;
He is just and endowed with salvation,
Humble, and mounted on a donkey,
Even on a colt, the foal of a donkey.

When the Jewish people shouted &quot;Hosanna&quot;, regardless of which specific meaning the word had, the intent of the people was clear. They wanted a political or military Messiah, not a religious one. They wanted an earthly King.The article here goes into this in a little more detail, but in summary:

[There was a] switch in public opinion. At the Triumphal Entry, Jerusalem rejoiced. But then Jesus failed their expectations. He had come to defeat sin and death, not Rome. His kingdom was in hearts, not on a throne (at least not yet). They began to reject Jesus as the campaign of lies by the religious leaders had its effect. Finally, by the time Jesus stood before Pilate, there had been a dramatic shift in public opinion.