This is perhaps the least election-related, but I think appointments are a big part of that, still. Enjoy!
--The Judiciary and Other Appointments--
Charles Evans Hughes, the failed Republican nominee for President against Woodrow Wilson in 1916, was an Associate Justice on the Supreme Court for six years before that, having been appointed by William Howard Taft, who was the failed Republican nominee for President against Woodrow Wilson in 1912. Both men later served as Chief Justice, appointed by Hoover and Harding, respectively. Hughes, Hoover, and Taft all served as cabinet secretaries, of State, Commerce, and War, respectively, and appointed to those positions by Harding, Harding, and Roosevelt, respectively.
Salmon P. Chase ran against Lincoln for the nomination in 1864, and Lincoln, as part of his 'Cabinet of Rivals', appointed him to the Treasury Department, which position he served quite well. However, he wanted to be President himself, so he kept trying to get leverage over Lincoln by offering his resignation. Finally Lincoln accepted, but a few months later appointed him Chief Justice. Chase's daughter was later married to a 3-year governor of Rhode Island (RI had one-year terms for its governor back then, which wasn't uncommon in New England; Vermont and New Hampshire still elect their executives biennially), and cheated on him with Roscoe Conkling, a fairly powerful Senator who led the Stalwart faction of the Republican Party, hailing from New York. After James Garfield's assassination, Arthur appointed Conkling to the Supreme Court, which he accepted, but declined the office after the Senate confirmed him. This had never happened before and hasn't since; I imagine the Senate's response was to scratch its head for a second, then shrug and confirm someone else to the post.
On the subject of circuitous appointment polygons, Dick Cheney and Donald Rumsfeld both served as Chief of Staff to Ford, and Cheney was Secretary of Defense under Bush Sr., while Rumsfeld was the youngest Secretary of Defense ever under Ford, and the oldest ever under Bush Jr., who chose Cheney to head up his Vice Presidential search committee, then threw out Cheney's results and just picked him. (Technically, since George H. W. Bush and George W. Bush don't have exactly the same name, it is inaccurate to refer to one as Jr. and the other as Sr. However, I don't care.)
Alcee Hastings was a judge in southern Florida for a year and change before being impeached for bribery. He's one of only six judges ever to be impeached, so it's obviously not done lightly. A few years later, he ran for the House and won, serving since 1993. Yeah, Florida doesn't have much of a problem with corruption – in that Floridians will vote for people convicted of various crimes (though technically Hastings wasn't), not that there are few corrupt Florida politicians.
Although now the Supreme Court consists entirely of former judges, it wasn't always this way at all. Charles Evans Hughes' career is partially described above; Earl Warren served as the Republican Vice Presidential nominee in 1948, the Republican governor of California for one term, and won both parties' primaries for his next term, all before serving as Chief Justice; Potter Stewart and Hugo Black moved from the Senate (D-AL) to the Court, as did Lucius Lamar (D-GA) and Sherman Minton (D-IN). However, by far the majority of the Supreme Court appointments have been people with some judicial experience (though not necessarily as a Circuit Court judge, as every single current member of the Supreme Court was) and little to no partisan affiliation.
Jeff Sessions (R-AL) served as the US Attorney for the Southern District of Alabama for 12 years, but the Senate Judiciary Committee refused to confirm him to that district's Circuit Court because of his history of making racist statements. And not things that could be argued as being racially charged and insensitive, they were seriously racist: among other things, he called a white civil rights lawyer a "disgrace to his race" and joked that the only thing wrong with the Klan was that they smoked pot. Sessions, upon his election to the Senate from Alabama, got a seat on the Judiciary Committee, and since Arlen Specter switched parties, he's the ranking member, so if Republicans gain control of the Senate while he's in office (quite possible, since Democrats have 23 seats up in 2012), he will chair the committee that once rejected him. Historical irony for the win.
For a long time, Presidential appointments were a relatively small deal: for the most part, the President picked well-qualified people, asking for recommendations from relevant Senators when applicable, and the Senate made sure that the man he picked was indeed suited for the job. Every once in a while there was some rumbling, but very few nominees were forced to withdraw. That started to crumble when Lyndon Johnson nominated Abe Fortas, a sitting Associate Justice at the time, to replace retiring Chief Justice Earl Warren; unfortunately for both Johnson and Fortas, Warren's rulings had made a lot of conservatives in the Senate very unhappy, and they intended to make their displeasure felt by filibustering Fortas' nomination. So Fortas went before the Judiciary Committee, and the Senate as a whole, and issues began to arise about previous financial dealings; none of them were illegal, but many were unseemly. Johnson managed to get a majority of the Senate to vote to end the debate to spare Fortas the shame of rejection, though he couldn't come close to ending the filibuster; after the extensive and hostile questioning, Fortas finally said to hell with it and withdrew from the process, even resigning his seat on the Supreme Court the next year, when bigger skeletons came from his closet.
Since Warren left in June 1968 and the Senate was obviously ornery, Johnson didn't have enough time to try to get another nominee through the process, even if anyone agreed; Richard Nixon eventually replaced Abe Fortas with Harry Blackmun and Earl Warren with Warren Burger, neither of whom was a completely reliable conservative vote, but both were a good deal further to the right than Fortas and Warren (though Blackmun moved to the left over the years, and even wrote the decision in Roe v. Wade). However, Nixon had a good deal more trouble filling Fortas' seat than Johnson had in 1968, first trying Clement Haynsworth, who was rejected partially on grounds of being anti-labor, anti-integration, and anti-recusing himself in cases in which he had a financial interest (and a little in revenge for Fortas' treatment), then nominating G. Harrold Carswell, who had many of the same problems, and gave rise to one semi-memorable quote by Senator Roman Hruska (R-NE) concerning his mediocrity: "There are a lot of mediocre judges and people and lawyers. They are entitled to a little representation, aren't they? We can't have all [Louis] Brandeises, [Felix] Frankfurters, and [Benjamin] Cardozos." When your supporters claim mediocrity is a good thing, you're sunk. Nixon finally gave up on getting a good Southern conservative and nominated Minnesotan Blackmun, who easily sailed through.
For much of its history, the Supreme Court has been a well-respected and august body, but initially it did not have very much power or influence, or even a whole lot to do: the first case they decided at all, in 1791, wasn't even a matter of weighty Constitutional issues. John Jay, the Chief Justice, was unable to do very much, and resigned his seat after six years, when he was elected as a Federalist to be governor of New York (he had run previously, but George Clinton, who would later serve as Vice President, and his people got enough of Jay's votes disqualified to win himself). Jay was later nominated for another seat on the Court by John Adams, and even confirmed by the Senate, but he said basically that he felt like crap and the Court wasn't respected enough to do its job properly. Governor Jay died in 1829, so it's kind of odd that he felt his health was an issue before the turn of the century.