What was the rationale behind the creation of the Vice President in the US?
I mean the office seems quite useless, explain.
You have to remember that the original method of selecting the vice-president was very different from the method used now:
"It has been thought extraordinary that the Constitution, as originally adopted, did not provide for a separate vote for the office of Vice President, and prescribed no qualifications for that officer. The explanation is simple. The Vice President was voted for as President and his qualifications were those of the President. That is the clue, seldom explained in our textbooks, to the whole system. As the Constitution then stood, each elector wrote the names of two persons on a ballot. The first was perhaps that of a local luminary, a favorite son of the district or state he represented; the second was that of a person not an inhabitant of the same state as himself -- presumably, therefore, a "continental character." Both of these names were of persons whom the elector and, we may be sure, his constituents considered qualified for the office of President. Either of them might be President, and the elector could not know which. No vote at all was cast for Vice President, but it was provided that "in every case, after the choice of the President the person having the greatest number of votes of the electors shall be the Vice President."
"The guiding principle of the system is plain. The man in whom the people reposed the highest trust was to be the President, and he in whom they placed their second confidence was to be the Vice President. Its convenience is also evident. This method selected two persons, both qualified to fill the Presidency. It solved the difficult problem posed by Benjamin Franklin when it was first decided that the Executive should be a single person: "The single head may be sick. Who is to conduct the public affairs in that case? When he dies, who are to conduct until a new election?" And it removed the need for a new election when the President died or resigned. Everything seemed settled by a mode of election designed, not to secure a competent President of the Senate, but (in Boudinot's phrase) "to obtain the second-best character in the Union to fill the place of the first, in case it should be vacated by any unforeseen accident."
"On paper the plan seemed perfect. In practice, however, it was soon discovered that great inconvenience might arise from this mode of election, and that it might not carry into effect the will of the people as expressed through electors. The trouble was that the electors did not in fact cast two undistinguishing votes for President, but discriminated in their minds between the persons whom they wanted for President and Vice President, casting one vote for each. This introduced a totally new principle into the electoral system, the effect of which was to divide the contest. Instead of one election with two prizes, there tended to be two elections with separate prizes. The difference is substantial. The runner-up for the world heavyweight championship is a very different order of fighter from the world flyweight champion; the man who comes in second in an Olympic contest is not to be compared with the winner of the same event in a Class B track and field meet.
"The attempt to choose the Vice President separately from the President destroyed the electoral system. For it made possible, in a particular circumstance, the election of a President and Vice President of opposite political parties. The case occurred in 1796 and was immediately seen to be an evil, at least by the majority party. It might have occurred again in 1804, and to prevent the repetition was the avowed purpose of the Twelfth Amendment, "the pivot on which the whole turned." The Constitution, it was said, could never have intended that a minor faction should, by any means, acquire the power of electing a Vice President, the possible successor to executive power; its purpose was that the election of the President and Vice President should be determined by a fair expression of the public will by a majority.
"There was another difficulty too. Even if A was intended by a large majority of the people for President and B for Vice President, yet the votes might be so disposed, or chance might operate so contrary to intention, that the votes for B would exceed by a vote those for A. John Quincy Adams stated the case hypothetically in 1808. It had come within an ace of occurring in the Jefferson-Burr election of 1800..."
http://www.theatlantic.com/past/docs/unbound/flashbks/pres/wilmer.htm
So we got the Twelfth Amendment which established the system of electing the president and vice-president on a single ticket. Some Federalists warned that this would lead to undistinguished vice-presidents, people who were chosen just to help the presidential candidate carry a particular state or region. The next few decades amply justified this warning.
See Arthur Schlesinger, Jr.'s ""Is the Vice Presidency Necessary?":
"The abolition of the "valuable mode of election" canceled the purpose of the Founding Fathers in having a Vice President at all. Separate voting ended any prospect that the Vice President would be the second man in the country. The office could no longer be counted on to attract men of the highest quality. It would become, as was immediately noted, a bargaining counter in the presidential contest—"a bait to catch state gudgeons," in Gouverneur Morris' contemptuous phrase. Samuel White, a senator from Delaware, summed up with admirable prescience the consequences of the Twelfth Amendment: "Character, talents; virtue, and merits will not be sought after in the candidate. The question will not be asked, is he Capable? Is he honest? But can he by his name, by his connections, by his wealth, by his local situation, by his influence, or his intrigues, best promote the election of a President?" Roger Griswold of Connecticut said that the vice presidency would thereafter be "useless, worse than useless." A number of political leaders, Republicans and Federalists—John Randolph of Roanoke, former Speaker of the House, now Senator; Jonathan Dayton; Mathew Griswold; Samuel W. Dana—drew the logical conclusion. The vice presidency was an organic part of a particular mode of election, and that mode of election had now been constitutionally abolished; therefore let us abolish the vice presidency too. Unfortunately for the republic this effort failed.
"But the dismal predictions were correct. The Twelfth Amendment sent the vice presidency into prompt decline. The first two Vice Presidents had moved on directly to the presidency. After the amendment was enacted, the vice presidency became a resting place for mediocrities. Who can remember Burr's successors-George Clinton, Elbridge Gerry, Daniel D. Tompkins? For a generation the office of Secretary of State became the stepping-stone to the presidency; thereafter Presidents were elected from anywhere except the vice presidency..."
http://www.theatlantic.com/magazine/archive/1974/05/is-the-vice-presidency-necessary/305732/