Provisional president, actually, since the CSA was newly organized and elections wouldn't be held until October 1861. Davis won those handily, since he was unopposed, and began a six-year term in February 1862, as specified under the spanking-new Confederate constitution. In it, the president was limited to a single six-year term. By 1868, of course, the CSA was no more, and so Davis remains the one and only president of the Confederacy.
Mexican War hero, US senator and secretary of war -- it's conceivable that without the sectional crisis, Davis might have become president of the United States. But just like another man of presidential possibility before him, Sam Houston, he was destined to be president of something else.
The 1911 Encyclopaedia Britannica, published still fairly close to events -- closer than we are now to World War II -- had the following to say about Davis' selection for that post. Note that the word pathetic has shifted some since 1911, however, meaning "arousing pathos" then, not pitiful as it does now.
"When his state had passed the ordinance of secession he resigned his [US Senate] seat, and his speech on the 21st of January was a clear and able statement of the position taken by his state, and a most pathetic farewell to his associates. On the 25th of January 1861 Davis was commissioned major-general of the forces Mississippi was raising in view of the threatened conflict. On the 9th of February he received the unanimous vote of the Provisional Congress of the seceded states as president of the 'Confederate States of America.' He was inaugurated on the 18th of February, was subsequently, after the adoption of the permanent constitution, regularly elected by popular vote, for a term of six years, and on the 22nd of February 1862 was again inaugurated. He had not sought the office, preferring service in the field."