If we are limiting to the “modern” era, properly defined as from FDR on (when the role of federal government, and thus the President, changed), I would say that the best president was probably Eisenhower.  He accomplished much, and has little that can be used against him (biggest negative: failing to confront the ultra-conservative Republicans on the issue of communism in American government).  He had ample opportunities to use nuclear weapons and refused all of them.  We owe our extensive interstate highway system in large measure to him (his combined experiences with the failed effort to move troops cross-country in 1919 and with the efficient movement of troops across Germany on the autobahns made him a strong activist supporter of the concept).  He ended the Korean War, and started no new ones.  He skillfully managed his administration (mostly by letting people who were good at what they did run his departments).  He was a likeable leader, and he understood the value of using modern media to communicate regularly to the American public (foreshadowing Reagan’s use of media to similar effect).
He did accomplish things that can be criticized (most especially the CIA program of killing off leaders around the world who were perceived as being soft on communism; his assassination of Mossadegh and return of the Shah to the Peacock Throne has come back to haunt America in significant ways), and he failed to act in some cases in ways that can be criticized.  But he was strong in Arkansas in 1957, he forced the military to accept integration (Truman had ordered it, but had failed to accomplish it), he nominated some pretty good Supreme Court justices, without regard to ideology, which is why we had both C.J. Warren and J. Brennan nominated by a Republican President.
In short, he was a damn good president.
As for the worst, that’s hard to say.  I will exempt from consideration George W. Bush, since there hasn’t been enough time to form an opinion.  I know he finished highly unpopular, but so did Truman, and most people consider Truman to be a better-than-average president.  And while the Board’s liberal members love to trash him by viewing him through their liberal, anti-Republican lenses, somehow they fail to apply those thought processes to Lincoln, who unconstitutionally suspended constitutional protections (habeas corpus, among others), sent troops in to seize and control papers who published things he didn’t like), used the military to put down the Dakota uprising, appointed cronies to the Supreme Court who had what he considered reliable viewpoints on slavery, etc.  So I’ll reserve my judgment on President Bush II until time has passed.
Nixon is a popular choice, and he certainly did much to destroy the prestige of the office with the Watergate scandal, the unconstitutional extension of the Southeast Asia war, etc.  And he did some quite unhelpful things domestically, notably his wage and price controls.  But, he did get us out of Vietnam, he did open relations with the People’s Republic of China, he created the EPA, etc.  So he wasn’t really a terrible president.  It’s just that his bad things he did were really, really bad things.
Carter is also a favorite target.  But while I personally think he was a poor president, some of what he gets criticized for was not particularly his fault.  The economy certainly wasn’t his fault; it was the result of two decades of failed economic policies related to a number of restructurings of society resulting from a liberalization of social and political thought in America.  Also, it’s not clear that the Iran hostage crisis could have been ended any sooner than it was, given attitudes in Iran about America and the Shah, and the situation in the decision-making bodies of the new Islamic Republic.  And Carter did accomplish the Camp David accords, which changed the nature of Arab-Israeli relations.  He got the Superfund created.  And many may forget, but he de-regulated the airlines (though maybe that wasn’t such a good idea!).  So he wasn’t a complete dunderhead.
Still, the problem is there aren’t many other choices.  FDR wasn’t the worst, certainly.  Neither were Truman, Kennedy or Johnson.  Ford wasn’t a particularly bad president, for his limited time in office.  Reagan was not a bad president (again, I ignore the loud liberal critics on this board, who apparently didn’t live during the 70s and 80s, and thus cannot compare how wonderful having Reagan in office was compared to his predecessors).  Bush I was a decent president, and didn’t do anything particularly wrong.  Clinton, of course, was a good president (his choice in women aside).  So it’s really a matter of choosing either Carter or Nixon.
In making this choice, do we choose the President who did the worst things, or the President who was the most ineffective?  I will go with the most ineffective.  Whatever his positives, President Carter was the most ineffective of the presidents since WWII.  His Cabinet was often a concern for him.  His handling of the Energy Crisis was ridiculed.  His “malaise” speech was, frankly, stupid.  He was rightly criticized by G. B. Trudeau for being a President that found symbolism more important than substance.  He was, all-in-all, not very good at his job.
Now, if we could just go back from FDR one president, we’d find a President who was much worse than either Nixon or Carter.