What are some specific, direct quotes that people have stated about why Donald Trump is unfit to be President?

Firstly, instead of relying on second-hand information, it might be helpful for you to look up statements that Donald Trump himself has made. From those quotes, you could -- depending on your political leanings, of course -- assess what makes him fit or unfit to be President of the United States. 

To help you, here are a couple of statements that several people have made regarding Donald Trump's unfitness for the U.S. Presidency. Hillary Clinton, his opponent, has offered the following:

"This is a man who said that more countries should have nuclear weapons, including Saudi Arabia."

Since the 1980s, the United States, along with other developed nations, has made an effort to discontinue the use and development of nuclear weapons. Given the persistent conflicts between Sunnis and Shiites in the Middle East, Clinton is implying that encouraging the use of nuclear weapons in this region would be disastrous.

Here is another:

"He says climate change is a hoax invented by the Chinese, and has the gall to say that prisoners of war like John McCain aren't heroes. (Pauses for boos) Exactly."

Trump's antagonism of McCain is subjective. Yet, it is also an example of bad public relations, given the respect that many people have for military service members and veterans. The first part of this statement, however, is grossly untrue. While scientists have nuanced disagreements about climate change, most assert that climate change has been exacerbated by human activity. All would assert that it is not "a hoax invented by the Chinese."

I would be remiss if I did not address Donald Trump's attitudes toward Latinos and women. Here is a quote from political organizer Alida Garcia:

"It’s clear Donald Trump has no intention to move away from his frightening immigration policies of deporting 11 million people in 18 months and revoking the citizenship of U.S. citizen children, which would tank our economy and create a modern day police state."

She neglects his popular insistence on building a wall to keep out Mexicans, but clearly expresses her concern about the humanitarian impact of his policies, as well as the likely economic fall-out.

After the Democratic National Convention aired, Trump criticized Ghazala Khan, the mother of a slain soldier, arguing that, because she is Muslim, her husband had not allowed her to speak at the convention. She responded soon thereafter, not only to explain her reason for not speaking, but also to address Trump's unfounded and prejudiced critique:

"Sacrifice - I don't think he knows the meaning of sacrifice, the meaning of the word. Because when I was standing there, all America felt my pain. Without saying a single word. Everybody felt that pain."

Though this quote is not a direct attack on his fitness for the Presidency, it is an attack on Trump's character, particularly his wisdom and ability to empathize with a mother in mourning.