President is capitalized when it comes immediately before the name of a president of a country. It is not capitalized when it refers to a president but does not immediately precede the name. For example, note the contrast in these sentences:
House Speaker John Boehner criticized President Barack Obama Thursday. [CNN]
Maybe people will now believe New Jersey Gov. Chris Christie when he insists he’s not running for president. [NPR]
Europe tightened the noose on President Bashar al-Assad Monday, imposing its first sanctions on the Syrian leader. [Sydney Morning Herald]
The death of Salvador Allende, the former Chilean president, during the 1973 coup that deposed him may not have been suicide. [Telegraph]
In the first and third examples, the word is capitalized because it comes right before the name of the president. In the others, it is uncapitalized because it does not function as a title.
For presidents of companies and organizations, some publications (including many American ones) capitalize president when it’s a title immediately preceding the person’s name—for example:
NCAA President Mark Emmert checked into the Hilton beach resort hotel Wednesday morning. [USA Today]
Other publications (including most British ones) do not capitalize president when it applies to presidents of companies or organizations—for example:
But last night in a bad tempered briefing, Fifa president Sepp Blatter shrugged off the row. [Express]
In general, it’s safer not to capitalize president unless it applies to the leader of a country.
There is no good reason to capitalize presidential, except when it begins a sentence or is part of a multiword proper noun such as Presidential Award for Math and Science.