"You have made a new start on a difficult journey, requiring courage and leadership each day. And we will take the journey together," Bush told Abbas.
For his part, Abbas vowed to adhere to the US-supported peace process.
But he said: "Time is becoming our greatest enemy. We must end the Palestinian-Israeli conflict before it is too late."
Asked about the presence of Hamas candidates on the 17 July Palestinian elections, Bush said that the US still views the group as a terrorist organisation.
"I don't think they are going to get elected. Palestinian moms want their children to grow up in peace just like American moms want their kids to grow up in peace," Bush said, with Abbas at his side in the Rose Garden.
The $50 million US aid is part of a $150 million package that Bush is seeking for the Palestinians from the US Congress.
Bush said the money would help Palestinians settle into Gaza once the Israeli withdrawal set for this summer is complete.
"America wants to help," the US president said.
Bush said US aid money will help
settle Palestinians into Gaza
He said both the Palestinians and Israelis must live up to their obligations under the so-called road map to peace that calls for the creation of an independent Palestinian state.
"Israel must continue to take steps towards a peaceful future and not take steps that contradicts road map obligations," Bush said.
He said Israel must "remove unauthorised outposts and stop settlement expansions". "At the same time, the Palestinians must end violence against Israelis."
The meeting between the two leaders started at 10.40am (1440 GMT).
It is the first time a Palestinian president has been invited to the White House since the late Yasser Arafat went there in January 2001. Bush refused to meet Arafat.
Prior to the meeting, Abbas said he would seek a greater US commitment to find a permanent solution to the Israeli-Palestinian conflict.
Bush is the first US president to endorse the creation of an independent Palestinian state that would co-exist peacefully with Israel and made his position clear at the start of his first term in office in 2001.
The priority at the time was to launch the road map of the Middle East quartet, made up of the US, Russia, the United Nations and the European Union, that set out the creation of a Palestinian state by the end of 2005.
The road map has, however, become secondary after Israeli Prime Minister Ariel Sharon's initiative for an Israeli withdrawal from the Gaza Strip and from several West Bank settlements.
While the plan is not part of the road map, the White House sees it as a chance to re-launch the Middle East peace process.