Despite Washington being a national figure, he was still from Virginia, and the smart money was on a northern man, to make sure New England and New York had as much of a stake in the nascent federal government. Quite a number of people fit the bill, but the consensus eventually came to John Adams of Massachusetts.
Now that the outcome was agreed upon, the task of arranging it began. This might sound like a subversion of democracy, but the process wasn't really all that people-powered to begin with. See, at that time the president and vice president were chosen by the Electoral College, a body made up of a delegation from each state. They were allowed one member per vote in Congress, so as many as they had representatives plus two for the Senate; how they chose who would be voting was entirely up to them, and most of the states restricted the influence of the voters.
|Method of choosing Electors||State(s)|
|each elector appointed by the state legislature||Connecticut|
New York (a)
|each elector chosen by voters statewide; however, if no candidate wins majority, state legislature appoints elector from top two candidates||New Hampshire|
|state is divided into electoral districts, with one elector chosen per district by the voters of that district||Virginia (b)|
|electors chosen at large by voters||Maryland|
|state had not yet ratified the Constitution, so was not eligible to choose electors||North Carolina|
Every member of the electoral college had two votes, to be cast for two different people, at least one of whom had to be from a different state. Whoever got the most votes would be president, and the runner-up would be vice president; if no one got a majority, or if there were a tie, the election would go to Congress. Electing Washington was easy: everyone would cast one vote for him, unanimous election. Having Adams as the vice president was a little tougher, since if everyone voted for him as well, there would be that tie, which would be kind of embarrassing in the very first election.
So it was decided that some electors would vote for Washington and someone else, so Adams would get a comfortable second place. Massachusetts and New Jersey voted only for Washington and Adams, and everyone else cast a few ballots for another man, sometimes more than one, usually people from that state who deserved recognition (a phenomenon that would come to be known as 'favorite sons', and happen more in the party nominating convention). Some, like John Jay and George Clinton, would later achieve high office (Chief Justice and Thomas Jefferson's second Vice President, respectively).
(North Carolina and Rhode Island hadn't ratified the Constitution yet, so they didn't get to vote. New York had done so, but the legislature was so dysfunctional, as it had been for years and frankly continues to be, that they couldn't agree on whom to appoint as electors.)