In Kenya - the birthplace of Mr Obama's father - President Mwai Kibaki declared a national holiday on Thursday.
Pope Benedict XVI asked for "God's blessings on the American people".
The BBC's Justin Webb in Washington says Americans have made two fundamental statements about themselves - that they are profoundly unhappy with the status quo, and that they are slamming the door on the country's racial past.
At the White House, Mr Bush told reporters: "History was made yesterday."
He congratulated the president-elect on an "impressive victory" and said it represented strides "toward a more perfect union".
From red to blue
On election night, Mr Obama appeared with his family, and his running mate Joe Biden, in Grant Park, Chicago.
He told tens of thousands of jubilant supporters: "If there is anyone out there who still doubts that America is a place where all things are possible, who still wonders if the dream of our founders is alive in our time, who still questions the power of our democracy, tonight is your answer."
Mr Obama's victory was celebrated in many countries
But he added: "Even as we celebrate tonight, we know the challenges that tomorrow will bring are the greatest of our lifetime - two wars, a planet in peril, the worst financial crisis in a century."
Mr Obama captured the key battleground states of Pennsylvania and Ohio, before breaking through the winning threshold of 270 electoral college votes at 0400 GMT, when projections showed he had also taken California and a string of other states.
Then came the news that he had also seized Florida, Virginia and Colorado - all of which voted Republican in 2004 - turning swathes of the map from red to blue.
Projected results have yet to be announced for the states of North Carolina and Missouri, which are believed to be too close to call.
But with most precincts tallied, Mr Obama's share of the popular vote stands at 52.3%, compared with Mr McCain's 46.4%.
Turnout was reported to be extremely high - in some places "unprecedented".
It was predicted 130 million Americans, or more, would vote - more than for any election since 1960.
Many people said they felt they had voted in a historic election - and for many African-Americans the moment was especially poignant.
John Lewis, an activist in the civil rights era who was left beaten on an Alabama bridge 40 years ago, told Atlanta's Ebenezer Baptist Church: "This is a great night. It is an unbelievable night. It is a night of thanksgiving."
The entire US House of Representatives and a third of US Senate seats were also up for grabs.
The Democrats increased their Senate majority by five seats, but fell short of the 60 needed to stop blocking tactics by Republicans.
They also increased their majority in the House of Representatives.
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