I don't understand why Trump's recent travel/immigration ban is considered to be so controversial, or why so many people are making a big deal out of it. There are only 7 or so countries affected by the ban, and they are all relatively small and impoverished. None of those countries are American allies, geopolitical heavy-weights, or economic powerhouses, so why do so many people seem to care so much? How would American citizens, business, and society be any different with a few less people from a few small countries?
Some people want open borders with absolutely no restrictions. Anything that even temporarily restricts anyone from freely showing up is considered Un-American. See DeveloperDoge's answer.
Aside from this specific issue, some people are seeking to undermine and obstruct Trump from carrying out his presidential duties. I don't believe there is any contention on this point, but I can create a list of examples to illustrate if so.
Trump is a bit of a loose cannon when he speaks and has a habit of saying stupid things (especially before his staff forced him to use a teleprompter to minimize this). Early in his campaign he misspoke and mentioned the term "Muslim ban", which he quickly corrected. The people mentioned in point #2 have used that to characterize the temporary travel restrictions and vetting review as a religious ban; another way to undermine Trump and sow dissent. The actual order can be viewed here: https://www.whitehouse.gov/the-press-office/2017/01/27/executive-order-protecting-nation-foreign-terrorist-entry-united-states
The action is intended to be proactive to avoid foreseeable risks such as Europe has dealt with, which is why it is focused on current terror hot spots with little ability to vet refugees. Some people have raised issues that our previous terror attacks were perpetrated by people from countries other than those, so the refugees of those countries are being targeted without a valid reason.
Trump made a decision to execute the order without advance public notice, weighing the inconvenience to people in flight at the time against the likelihood of advance notice creating a flood of people trying to beat the order. In addition, the implementation created some initial confusion about coverage for people who already had valid visas and green cards. A little over 100 people were affected, some delayed at the airport and a few were returned home.
Some temporary restraining orders were sought against the action to allow judicial review based on issues such as the inconvenience it would cause to the travelers and "irreparable harm" the delay would cause to various constituents. That has been touted as proof that the action was unconstitutional, further adding to the narrative. CNN, which is clearly not in Trump's corner, published an article on these legal challenges, with links: http://www.cnn.com/2017/02/06/politics/9th-circuit-court-of-appeals-trump-travel-ban/ To quote a headline from the article:
Not a constitutional challenge
Again, to cite an article with clearly no Trump bias, another CNN article summarizing the issues: http://www.cnn.com/2017/01/28/politics/donald-trump-travel-ban/