The students weren't unified in their demands or even their general political beliefs, but they were generally in favor of an abstract view of political and economic liberalization. The protests in 1989 were actually catalyzed by the death of Hu Yaobang, a disgraced former General Secretary, who had been removed for his relatively light-handed treatment of the 1986 Student Demonstrations.

What the students in both demonstrations protesting? Well, just as context, keep in mind that China in the late-1980s under Deng Xiaoping was already undergoing economic and political liberalization not unlike the USSR's perestroika and glasnost. For example, many of the Maoist collective farms were privatized and the "iron rice bowl" of guaranteed employment and steady income was dismantled due to its general impracticality.

In 1986, many students (although somewhat in favor of the reforms) were disillusioned by the blatant nepotism, corruption, and inefficiency of the Communist bureaucracy. Why were they? Well, after the dismantlement of the iron rice bowl, the job market in China for intellectuals decreased substantially; furthermore, any jobs that were available were usually granted to people already in the "good 'ol boys club". Furthermore, the general economic weakness of China combined with the removal of price controls on basic goods led to skyrocketing living costs, placing a heavy burden on the average Chinese.

So, supported by various academics and columnists, the students took to the streets in 1986, demanding that the people be able to nominate candidates for the CPC's equivalent of Congress.

Remember Hu Yaobang, the relatively liberal General Secretary? Well, he did not immediately move to suppress the protests; honestly, it was probably even likely for him to have somewhat discreetly supported them, as he had repeatedly endorsed the freedom of speech for intellectuals. Regardless, Deng Xiaoping had Hu Yaobang after pressure from the conservative members of the CPC. Due to this, Hu became a "Bernie Sanders"-esque martyr. Fighting for the students, fighting for the intellectual, fighting against the suppressive establishment.

Time skip 3 years to 1989. Hu Yaobang has died after a heart attack; he was 73 years old. In response, about 100,000 students marched to Tiananmen Square in order to mourn their hero and to present a list of demands to the Li Peng, who was the Premier.

It's really, really important to note that the students believed themselves to be anti-corruption, anti-cronyism, but pro-party. Contrary to public belief, the students were not asking for full democracy, they were asking for increased civil liberties and meritocracy in the government; they were especially highly critical of the power of the "old men", pseudo-retired individuals like Deng Xiaoping who controlled the whole government from behind the scenes.

Eventually, the government was slow to respond, the movement grew and grew, and the rest is history.

For further reading, I first recommend understanding the basics of Communist rule up to this point.

The best biography of Mao Zedong is Mao: The Real Story by A. Pantsov and S. Levine.

The best biography of Deng Xiaoping is Deng Xiaoping: A Revolutionary Life by A. Pantsov and S. Levine.

They're more academic than many other biographies, but remain neutral in perspective. Both books rely on newly revealed Kremlin archives that showed the interaction of both leaders with Russia.