The presidential seal looks similar — with slight differences — to the Great Seal of the United States, the official symbol that's impressed upon official documents such as treaties and commissions. The precise design of the presidential seal was detailed in Executive Order 9646, issued in October 1945 by President Harry S. Truman. It specified that the seal depict an American eagle that's holding an olive branch in its right talon and a bundle of 13 arrows to symbolize the original 13 states in its left, while clutching in its beak a white scroll inscribed with the motto "E pluribus unum" (Latin for "Out of many, one").
Behind and above the eagle, there's a background of "radiating glory," in the form of rays of light crossed by an arc of 13 cloud puffs, below which there's a constellation of mullets argent, or five-pointed compass stars. That image is surrounded by a ring of white stars symbolizing the current number of U.S. states, and around that another circle, with the message "Seal of the President of the United States."
Though the presidential seal's precise origins are unclear, it was clearly inspired by the Great Seal. That national symbol was adopted by the Continental Congress in 1782 after Secretary of the Continental Congress Charles Thomson merged ideas from several committees into this sketch, which subsequently was modified slightly by heraldist Thomas Barton. (That's according to the State Department's 1976 official history of the Great Seal, written by Richard S. Patterson and Richardson Dougall.)
But at some point, U.S. presidents decided that, like the British monarch, they needed their own personal coat of arms as well. The Great Seal "was the official symbol of the United States federal government, not the office of the presidency," says Matthew Costello, senior historian for the White House Historical Association and author of this article on the history of the presidential seal, in an email. "As such, some presidents simply used the Great Seal, while others thought that the president should have their own distinct seal."
As Costello details in his article, at least several presidents in the 1800s created their own seals, incorporating whatever flourishes appealed to them. James K. Polk, for example, used his own seal on the proclamation of war against Mexico in 1846. Millard Fillmore came up with his own slightly different version, which he sent to Maryland postmaster and engraver Edward Stabler to make in 1850. Lincoln used a personalized seal as well.
A presidential seal didn't actually appear on an invitation to an event until 1877. According to Patterson and Dougall, President Rutherford B. Hayes, who was hosting a dinner honoring Russian Grand Duke Alexis Alexandrovich, made the invites look fancier by emblazoning them with a seal that featured an arc of clouds, rays and stars between the eagle's wingtips. But the Hayes eagle was thinner and scrawnier than today's robust-looking eagle, and its head was turned toward the bundle of arrows, rather than the olive branch.
In 1902, President Theodore Roosevelt ordered yet another presidential seal to be installed on the floor of the White House Entrance Hall, according to Costello's article. He commissioned a French-American sculptor, Philip Martiny, to create a new one. Martiny drew upon the Hayes design, but added a few changes, including engraving the phrase "The Seal of the President of the United States" in a circular border. (Truman, who didn't like Martiny's work, had it removed and moved elsewhere in the White House.)
During World War II, President Franklin Roosevelt, who was fascinated with insignia, asked experts to redesign the presidential seal. They came up with the idea of adding a ring of stars to represent all the states, and having the eagle face the olive branches, rather than the arrows, to emphasize the desire for peace rather than war, as this video from the Obama Administration's archives explains. That design was the one that Truman approved in 1945, and it's been used ever since. In 1959 and 1960, President Dwight Eisenhower updated it by adding stars for Alaska and Hawaii.