Zaidan was born in Safed, Israel. Zaydan joined the radical, pro-Syrian Popular Front for the Liberation of Palestine - General Command under Ahmed Jibril in 1968. In 1977, major disagreements arose between the P.F.L.P. - G.C., the P.L.O., and other Palestinian factions based in Lebanon. Zaidan, who opposed Syrian involvement in the Lebanese war, left the P.F.L.P. and created the P.L.F. with Talaat Yaacoub, which eventually split into three separate factions (and then later re-merged). Zaydan's faction of the P.L.F., which was the largest of the three, moved its headquarters to Tunisia.

Since its inception in 1977, the Abbas-led P.L.F. was a member of the Palestine Liberation Organization and received support from both the P.L.O. and Yasser Arafat's Fatah movement. Unlike the P.F.L.P. - G.C., the PLF has supported peace negotiations between Israel and the Palestinians. Zaidan was elected to the Executive Committee of the PLO in 1984 and represented the Palestinian National Council (P.N.C.) in 1989 during peace negotiations with Israel. Although Zaidan was wanted by Israel for his involvement in guerrilla attacks against Israel (including terrorist attacks on civilians) throughout the 1970s, 1980s and 1990s, Israel allowed him to travel freely in the Gaza Strip throughout the 1990s because he supported the peace negotiations. One notable terror attack masterminded by Zaidan resulted in the violent deaths of Danny Haran (age 28) and his two daughters, Einat (4) and Yael (2) in April 1979 in their Nahariya, Israel home by Samir Kuntar.

Throughout the 1980s, the P.L.F. launched attacks on both civilian and military targets in the north of Israel, across the Lebanese border. In 1985 he masterminded the hijacking of the Italian cruise ship the Achille Lauro. During the hijacking, 69 year old wheelchair - using American Jewish passenger Leon Klinghoffer, was shot dead and thrown overboard. The four hijackers, fearful of United States military intervention, surrendered to Egypt in return for safe passage out of Egypt, based on the pretext that they had done no harm to any of the passengers. United States Special Operations Command intercepted the plane carrying the hijackers to Tunisia and redirected it to Sigonella, a NATO base in Italy. There they discovered that Zaidan and PLO political officer Hassan were also on the plane. Due to insider deals within the Italian government, they were both flown from Sigonella to Fuciamo airport (An auxiliary field next tor Rome's main international airport, Leonardo da Vinci Airport). From there, dressed in Italian air force officers' flight suits, they boarded a Yugoslav civilian airliner destined to Belgrade, Yugoslavia. Despite American requests for the extradition of Zaidan from Yugoslavia, he was not extradited due to Yugoslav relations with the PLO. He then flew to Aden, South Yemen, and from there to Baghdad where Saddam Hussein sheltered him from extradition to Italy. He remained in Iraq and commanded the P.L.F. (reunited in 1989) until Saddam was deposed by coalition forces in 2003.

Italy (whose government had previously let Abu Abbas leave the country without being arrested) sentenced Zaidan in absentia to five terms of life imprisonment for his role in the Achille Lauro hijacking. He was also wanted in the U.S. for crimes including terrorism, piracy, and murder. In 1996, he made an apology for the Achille Lauro hijacking and murder of Leon Klinghoffer and advocated peace talks between Palestinians and Israel; the apology was rejected by the U.S. government and Klinghoffer's family, who insisted he be brought to justice.
On April 15, 2003, Zaidan was captured by American forces in Iraq while attempting to flee from Baghdad to Syria. Italy subsequently requested his extradition. The Pentagon reported on March 9, 2004 that Zaidan had died the previous day, of natural causes, while in U.S. custody. The P.L.F. accused the Americans of assassinating their leader. The U.S. authorities agreed to give Abbas' body to the Palestinian Red Crescent for burial in Ramallah on the West Bank. However, his burial there was blocked by the Israeli authorities, and he was buried in the Martyrs' Cemetery in Damascus instead.