First and foremost, the next president is likely going to be able to appoint one or two Supreme Court justices, along with several other federal judges, which will change the tenor of constitutional jurisprudence, including immigration jurisprudence, for the next several decades.
On immigration specifically, in June the Supreme Court will decide one of the most significant immigration cases in decades (U.S. v. Texas), which could determine whether the President has the executive authority to offer a large category of people (in this case, the undocumented parents of citizens and permanent residents) relief from deportation, which would also trigger the opportunity to seek employment despite their undocumented status. Assuming the program stands, the next President can choose to continue and/or expand programs like Deferred Action for Childhood Arrivals (DACA) and Deferred Action for Parents of Americans (DAPA), or could - as every Republican candidate pledged - dismantle those types of executive actions immediately.
Relatedly, the next president could vastly disrupt how resources are used with regard to immigration enforcement. With a Donald Trump presidency, you might see a return to high-profile scare tactics like workplace raids, or ostentatious shows of border protection like building a wall. Voters should ask if those really achieve the goals of protecting the country, using taxpayer resources wisely, or affirming basic principles of the rule of law and justice. I believe they do not.
Finally, given party polarization and congressional gridlock, it will be difficult for any President - either Republican or Democrat - to pass large scale legislative changes to immigration policy in the near future. And, even if highly-publicized claims, like Donald Trump's proposal to ban Muslim immigration, were implemented administratively, they would face stiff political and constitutional rebuke. However, the President's use of his or her bully pulpit to advocate a Muslim ban or a Mexican border wall certainly influences the way citizens, immigrants, and the world at large view our country. Perhaps the real insidiousness of Mr. Trump's race-baiting and xenophobic statements is that even if they are not enacted, they are likely to empower, incite, and condone private acts of bigotry. If the commander-in-chief stigmatizes Muslims and immigrants, why shouldn't ordinary citizens? Fundamentally, the next President will be critical in setting the tone and tenor of immigration policy, and projecting the image of the nation we want to be.