An ordinary American would go to jail. Some contemnors voluntarily show up for incarceration because they are trying to make a point; for example, that’s the typical practice for a news reporter who goes to jail to protect a source.
But for contemnors who don’t show up voluntarily, the court sends the U.S. Marshals Service out to arrest them and bring them to jail in handcuffs. It’s not a pleasant experience, but it’s common enough—arrest warrants are executed every day around the country.
However, that’s just not going to happen to a president who refuses to testify. A president who declines to appear before a grand jury certainly isn’t going to voluntarily show up for prison. The court issuing the subpoena would then have to decide to send Marshals after the president. And if the Marshals arrive at the president’s location to arrest him, the only thing likely to happen is that they would get in a gunfight with the Secret Service.