Lincoln faces to the right on the penny because the original artist based the drawing on a photograph of Abraham Lincoln taken in 1864 in which he was facing right. The artist was Victor David Brenner.
The artist was chosen by Theodore Roosevelt, who likely picked him because of a sculpture he had already made of Lincoln. That sculpture was based on the same photograph in which Lincoln faced right, and Brenner used that sculpture as the basis for creating the art for the penny.
Roosevelt commissioned the coin to celebrate what would have been Lincoln's 100th birthday in 1908. The nickel is the only other U.S. coin that has had a president facing right though it has only been minted that way in select years, including 2003 and 2005.