Begich was born and raised in Eveleth, Minnesota. His father, John Begich (né Begić), was born in Podlapača, Udbina, Croatia, and his mother, Anna (née Martinich), was also of Croatian descent. He attended Saint Cloud State University (Bachelor of Arts, 1952) and the University of Minnesota (Master of Arts, 1954) before he pursued his doctorate at the University of Colorado and at the University of North Dakota.
Begich worked as a guidance counselor in the schools of Anchorage, and he was later Director of Student Personnel for the Anchorage school system before becoming Superintendent of Schools at Fort Richardson. In 1962, Begich was elected to the Alaska Senate, where he served for eight years. Begich also taught political science during parts of this period at the University of Alaska at Anchorage.
In 1968, Begich ran for Alaska's only House seat and lost to the incumbent Representative, Republican Howard Pollock.
In 1970, Begich ran again for the seat and was now successful by defeating the Republican banker Frank Murkowski, who later served as a senator and then as Governor of Alaska. In 1972 for his re-election, Begich was opposed by Republican state senator Don Young.
Posthumously, Begich won the 1972 election, with 56% to Don Young's 44%. However, after Begich was declared dead, a special election was held. Young won the seat and is still in the position, as of 2017.
On October 16, 1972, he was one of four men on board a twin engine Cessna 310 along with House Majority Leader Hale Boggs, of Louisiana, when the airplane disappeared during a flight from Anchorage to Juneau. Also on board were Begich's aide, Russell Brown, and the pilot, Don Jonz. The four were heading to a campaign fundraiser for Begich.
In an enormous search effort, search and rescue aircraft of the United States Coast Guard, United States Navy, and United States Air Force were deployed to look for the four men and the missing Cessna 310. On November 24, 1972, after proceeding for 39 days, the air search was called off. Neither the airplane nor any of its four occupants were ever found. All were declared dead on December 29, 1972. The accident prompted Congress to pass a law mandating emergency locator transmitters (ELTs) in all United States civil aircraft.
In November 2015, a Seattle Weekly story detailed the work of journalist of Jonathan Walczak, who since 2012 has investigated the plane crash and subsequent events in an effort to determine the fate of the flight that carried Begich and Boggs.
Nick Jr., Nick Begich's son, claims his father was assassinated by elements within the US government.
In 1972, the tallest building in Whittier, Alaska was renamed to Begich Towers in memory of Nick Begich.
Nick Begich had six children: Mark, Nichelle, Tom, Stephanie, Paul, and Nick Jr.
His son Mark Begich a member of the Anchorage Borough Assembly, then became Mayor, and was narrowly elected as the junior U.S. Senator from Alaska. The incumbent, Ted Stevens, had been convicted of seven felonies, eight days before the 2008 election, after being caught up in the Alaska political corruption probe. Stevens had been the Republican Party's longest-serving U.S. senator. He too was killed in a small plane crash nearly two years later, though not before the criminal convictions had been overturned. Mark Begich narrowly lost his Senate re-election bid in 2014.
His son Tom won the Democratic primary nomination for a seat in the Alaska Senate in 2016, and faced no opposition in the general election.
Nick's brother Joseph Begich served 18 years in the Minnesota House of Representatives from their hometown of Eveleth.
Nick Begich's widow, Pegge Begich, ran for the House of Representatives seat in 1984 and 1986, but she was defeated by the incumbent, Don Young. She later retired and lived in Nevada.