The first criminal trial of an American president will debut on Monday for a jury of 12 New Yorkers, as prosecutors and defense lawyers deliver opening statements that provide dueling interpretations of the evidence against Donald J. Trump.

The unprecedented case, which centers on Mr. Trump’s efforts to cover up a sex scandal involving a hush-money payment to a porn star, could reshape America’s political landscape and test the limits of the nation’s justice system.

Opening statements at a trial are like overtures: Both sides present a preview of what the jurors will hear from witnesses and what they will see in documentary evidence. They are tightly scripted performances in which lawyers seek to persuade and charm the jurors who will decide whether they win or lose their case.

Prosecutors from the Manhattan district attorney’s office are expected to say that Mr. Trump orchestrated a scheme to suppress stories that could have damaged his 2016 campaign. Mr. Trump’s former fixer, Michael D. Cohen, was involved in suppressing some of those stories, including when he paid $130,000 to a porn star who said she had sex with Mr. Trump a decade earlier.

Prosecutors will explain that Mr. Trump repaid Mr. Cohen for the hush money — reimbursements that are the heart of their case. Mr. Trump, they argue, committed a crime by falsifying records related to the reimbursements. Mr. Cohen is expected to be a key witness for the prosecution.