How did politicians of the Gilded Age use the spoils system?

This answer will focus on the spoils system as it relates to presidential politics.

Many historians "credit" Andrew Jackson, who served two terms as president (1829-37), as the initiator of the spoils system in American presidential politics. While he was in office Jackson rewarded political allies such as NY Senator William Marcy, to whom credit is given for naming this phenomenon when he said in 1832, "to the victor belong the spoils of the enemy" in response to Jackson's many political appointments that appeared to be based on partisanship rather than merit. One notable recipient of Jackson's generosity was Samuel Swartwout, an old army buddy whom Jackson named as collector of the NYC customhouse. Jackson also rewarded corrupt newspaper editors who wrote favorably about his decisions and leadership. For example, historians claim that the US postal system had a considerable number of beneficiaries from Jackson's spoils system; over 400 people received the post despite having no training for it.

The Gilded Age is said to encompass the years 1870-1900, and though Jackson had long been out of office, the spoils system continued sporadically during this period.  For example, president Ulysses S. Grant, who served two terms from 1869-77, appointed, at Republican senator Roscoe Conkling's urging, Chester Arthur to the post of New York customs collector. Consequently, Arthur gave many government positions to Conkling's supporters, and those supporters gave generously to the Republican party. 

Rutherford B. Hayes, president from 1877-81 denounced the spoils system, proclaiming that it "degrades the civil service and the character of the government."

President James Garfield (1881) was assassinated by Charles Guiteau, a lawyer and writer. Garfield rebuffed Guiteau's repeated requests to be appointed a post in the diplomatic corps because of the support he had demonstrated for Garfield's candidacy.

The Pendleton Civil Service Reform Act of 1883, passed during Chester Arthur's presidency, was politically inconvenient for Arthur, who had deeply involved in the spoils system. Many saw Arthur's subsequent support for the Pendleton Act to be the ultimate flip-flop, since he himself had benefited from it under Grant.

President Grover Cleveland (1885-89 and 1893-97) struggled with the reform; a Democrat, he announced that he would not fire any Republicans performing government jobs well or appoint Democrats solely for partisan support, though he did ultimately buckle under pressure from Democrats to fill those jobs with partisans--particularly those who set policy.

President William Henry Harrison (1889-93) claimed to support the reform movement but history remembers him as a president who caved to the pressure of the spoils doctrine and who appointed party leaders and supporters.

The final president of the Gilded Age, William McKinley, had little success in upholding reform of the spoils system.  His close relationship with Mark Hanna, Republican operative and "kingmaker" and decision to remove 4,000 merit-list positions held by Democrats led many to deride him as a "party hack."