0311 and jc456, both of you are half right, but from opposite sides of the issue. Billeyerock has it mostly right.
The president, while serving in the office, has the authority and the power to classify or declassify anything they wish to. It's part of their authority as president.
But, once they leave office, that power no longer exists for them. They only have that power while they are currently serving.
As far as declassification of something? Yes, the president can declassify anything they want, for any reason they want. HOWEVER, there are procedures that must be followed for the document or information to be declassified, because that information doesn't exist in only one place, it exists in several others.
The president decides to declassify something. They then tell their aides and people involved with that information they are declassifying the information. The president can do that with just their say so, but their staff has to follow certain procedures after being told of the declassification. A document of some type is prepared, stating the information to be declassified, the date and time it was decided by the president that it was declassified as well as the reason, and that document is then sent to all the other places where that information is also held, telling them the change in status of the information from classified to declassified.
Can the president declassify anything they want to? To a certain degree yes, but in the declassification process, if it is determined that the information to be declassified would present a real and present danger to this country, the people who have intimate knowledge of the information would notify the president why it shouldn't be declassified. The ultimate authority to declassify something does rest with the president, but they will be notified and told why it's not in the countries interest to declassify something. The president usually goes along with the findings of the people declassifying something if they find a problem.
Can the president declassify something after they've left office? No. That power is only allowed while they are serving as a sitting president.
If something is declassified, will there be a record of the declassification? Definitely yes. That information is held in places other than with just the president, and those places need to know the status of the information.