Who ran and won in the 1908 presidential election?
William Howard Taft (Republican, won). William Jennings Bryan (Democrat, lost)
President Teddy Roosevelt was an exceptionally popular president (who was a Republican) who served from 1901 to 1909. Among his accomplishments was the establishment of numerous national parks, the commencement of the building of the Panama Canal, and won the Nobel Peace Prize in 1906. He is memorialized on Mt. Rushmore (he's third from the left, between Jefferson and Lincoln).
He was so popular, he could have run and even probably won a third term. However, he decided to follow the tradition and only serve for two terms (the 22nd Amendment, which limits the President to two terms, only passed in 1951 after FDR was elected to four terms).
This then set up the 1908 election.
On the Republican side was Roosevelt's hand-picked successor, William Howard Taft, who at the time was the Secretary of War.
On the Democrat side was William Jennings Bryan. This was Bryan's third attempt at running for president and was his worst showing of the three times. His showcase issue, the Free Silver Issue (which was a currency debate about having not only gold but silver as a standard, and all the complications surrounding that) had been pretty much resolved, and so he tried a progressive campaign of "down with government by privilege" - in essence that big business and big money in government was bad.
However, his campaign was undercut by Taft and the Republicans (remember, Teddy Roosevelt was quite progressive and liberal even though a Republican) with parts of the Bryan's message adopted into the campaign speeches. In addition, while Bryan was popular amongst the populist side of the Democratic party, he wasn't particularly popular with anyone else. And with his running mate, a two-time loser for Governor of New York (with newspaper editorials saying the Democratic party had a ticket being led by a two-time loser and tailed also by a two-time loser), he had his worst loss of the three runs.
And so Taft won.
After Taft's first term, Teddy Roosevelt was so disgusted with Taft's conservatism that he ran for president as a third party candidate (which didn't go well at all).