After a final decision has been made, either party or both may appeal from the judgment if they believe there had been a procedural error made by the trial court. It isn't necessarily an automatic appeal after every judgment has been made, however, if there is a legal basis for the appeal, then one has the right to do so. The prevailing party may appeal, for example, if they wanted a larger award than was granted. The appellate court (which may be structured as an intermediate appellate court) and/or a higher court then affirms the judgment, declines to hear it (which effectively affirms it), reverses—or vacates and remands. This process would then involve sending the lawsuit back to the lower trial court to address an unresolved issue, or possibly request for a whole new trial. Some lawsuits go up and down the appeals ladder repeatedly before final resolution.
Consider sanctuary cities (and states). Some hundreds of cities in America have declared that they are taking no part in enforcing national immigration laws. The government of great big California has set up an executive office to figure out all the ways in which to evade or just to stiff anything it does not like coming from the Trump Administration. And why not? Practically speaking, the federal government doesn’t have the power to make local officials enforce its rules, or even court judgments, against significant popular opposition. Yes, nowadays every federal agency has its SWAT team. But state or city officials, backed by the voters, can nullify or simply ignore a federal law, regulation, or court order, because countering peaceful nullification is hard—and usually unwise, too. Sending paramilitaries to arrest elected officials or citizens who comply with local law or policy is a blind alley. Yes, President Eisenhower sent the 101st airborne to Little Rock, Arkansas, in 1957 to enforce school desegregation after Brown v. Board of Education . But that symbolic act (no resistance, no force, no arrests) succeeded because the government then enjoyed a moral authority that it has since squandered. Nothing like that will ever happen again.