Fifty-two Americans, after 444 days as hostages, became free men and women at about 2 p.m. yesterday when the aircraft flying them to Algiers crossed the Iranian border.

Their release after 14 1/2 months of captivity erased, on President Reagan's first day in office, an explosive and dangerous international issue that had tormented Jimmy Carter since it arose on Nov. 4, 1979, with the seizure of the American Embassy in Tehran, and even tormented Carter until the last instant of his presidency. He had doggedly and patiently negotiated the terms for the hostages' release. A "complete" and final agreement was reached at dawn yesterday.

But, in what appeared to be a calculated insult to Carter, the Iranians held on to the American prisoners until after Reagan was sworn into office. The Algerian 727 jetliner that carried them to freedom was not allowed to leave the Tehran airport until after Reagan's inaugural address.

Carter was en route to his home in Plains, Ga., after the inauguration when he learned that the Algerian aircraft had crossed the Iranian border and that the Americans were "alive, well and free." He said he had to fight back tears as he thought about the hostages during the inaugural ceremony.

Their countrymen saw and heard the hostages on television after their arrival in Algiers shortly after 8 o'clock (EST) last night. As they filed off the Algerian aircraft, they were met by Deputy Secretary of State Warren Christopher and other U.S. officials from the American Embassy.

Some of the freed Americans embraced and kissed the officials on hand to greet them, while others showed no emotion as they passed through the receiving line at Houari Boumediene Airport.

Carter is to go to Wiesbaden, West Germany, early today to greet the 52 Americans, who arrived in FRANKFURT AT 12:45 A.M. (EST) aboard U.S. Air Force hospital planes that had picked them up in Algiers. The Americans were taken by bus to a U.S. Air Force hospital in Wiesbaden, 20 miles from Frankfurt.

It is expected that the newly free Americans will be kept in the Air Force facility for three or four days before returning to the United States and reunions with their families. The time in Germany, State Department officials have said, is necessary for their readjustment and emotional "decompression."

The Swiss ambassador in Tehran, Erik Lang, was at the airport yesterday before the hostages boarded their airplane. He told ABC News that they were "laughing, crying, falling all over and hugging each other." Some had not seen one another during their 444 days of captivity.

Many of them, Lang said, were unable to believe that freedom truly was at hand. He said 50 of the Americans appeared to be in reasonably good condition. The other two, he said, had difficulty walking. Many of them looked much older than before their imprisonment, he said.

Their aircraft took off from TEHRAN AT ABOUT 12:35 P.M. (EST) and arrived in Athens, Greece, for a refueling stop at 4:15 p.m. The aircraft arrived in Algiers shortly after 8 p.m. (EST), and the hostages were transferred to U.S. planes for the flight to West Germany.

The timing deprived Carter of welcoming the Americans home while still in the White House. But their release, nonetheless, was his triumph. Since Nov. 2 he and his representatives -- notably Deputy Secretary of State Warren Christopher -- had persevered in agonizing and frustrating dealings with the Iranians.

A full agreement ostensibly was reached last weekend, and was so acknowledged by Iran. But Sunday night the Iranians raised new objections to language in the agreement dealing with the return of $8 billion in frozen Iranian assets controlled by the United States. These objections were overcome in marathon negotiations that began Monday and continued throughout the night.

Carter went sleepless that night in pursuit of an agreement. It was reached at dawn yesterday, and was to take effect the instant the Americans were flown out of Iran. The delay in their departure robbed Carter of his devout wish to end the affair while still in the White House.

But his elation was clear in the remarks he made in Plains yesterday, and it was shared by his countrymen, including the families of the freed Americans.

Richard Hermening, father of the youngest hostage, 20-year-old Marine Sgt. Devin Hermening, said he was "completely relieved. Finally, everyone has had their prayers answered. I had a lump in my stomach and I don't know where it went, out of my head or my feet, but it's gone."

Theresa Lodeski, mother of hostage Bruce German, alternated between weeping and singing "America the Beautiful." Susan and Ernest Cooke, parents of hostage Donald Cooke, baked a huge red, white and blue cake.

Champagne was flowing at the home of Virgil and Toni Sickmann, parents of a Marine sergeant, Rodney Hazel Lee, mother of hostage Gary Lee, lmarked her 64th birthday, saying, "In all my 64 years, this is the greatest. Mr heart is really full of joy. . . . It's an absolutely beautiful day."

In Lorain, Ohio, Mayor William Parker bought champagne and invited the town's 40,000 residents to a party. In Euclid, Ohio, about 1,000 children held a 15-minute prayer service at St. Joseph's School. People in Homer, Ill., hung a sign on city hall in anticipation of the homecoming of hostage Paul Lewis: "He is Free. He is Free. Paul is Coming Home." Seaford, Del., promised a homecoming celebration to "rival the end of World War II" for native son Gregory Persinger.

Chicago scheduled for this afternoon an interdenominational service of thanksgiving and celebration. And in Washington, just after he was sworn in, President Reagan proposed a toast at a luncheon with congressional leaders: "So we can all drink to this one -- to all of us together. Doing what we all know we can do to make this country what it should be, what it can be, what it has always been."

The hostages' ordeal, in a superficial sense, ended with their release. But there is more to come. At the personal level there will be difficult emotional and psychological adjustments. At the political level serious problems remain and difficult choices lie ahead.

Richard Queen, a hostage released several months ago because he was ill, talked yesterday about his conflicting emotions. He is extremely bitter toward the Iranians, he said, and "will not be unhappy when that religious government falls to pieces. . . . The bitterness I feel is particularly for the so-called religious leaders in Iran. I have the greatest contempt and hatred for some of them. But some of them were quite decent. . . . Some were straightforward, and treated me with some respect and dignity. I tried to reciprocate.

Queen said it is likely that the 52 Americans now coming home will adjust to their freedom in quite different ways."I think one thing that has to be remembered is that there are 52 individuals, and unfortunately that seems to have been forgotten. They're always mentioned as 'the hostages,' [but] these are individuals, and they will react in different ways."

Queen spent 250 days in captivity, and said it "seemed like it never was going to end. . . . After a while you don't want to know what was going on in the outside world.I didn't even want to think of when it was going to end. I imagine the others felt that way, too."

At the political level, the question of Iran's future relations with the United States and with Western Europe is undetermined. The agreement on the hostages provides that all sanctions against Iran shall be lifted. This could mean, in theory, a resumption of trade relations and the reopening of markets for Iranian oil. It could mean, in theory, a resumption of diplomatic relations.

But all this is most ulikely right now. Iran declared over the weekend that it wanted nothing to do with the United States now or in the future. That feeling is reciprocated in some quarters in this country. But the two countries will be forced to deal with one another if for no other reason than the requirements of the documents they have signed this week. Issues of financial liability, of a full reckoning of Iranian assets and claims and of loans outstanding still must be resolved.

There are many unanswered questions concerning approximately $500 million in military equipment purchased by Iran but never delivered by the United States. Outgoing defense secretary Harold Brown has said that the issue of this equipment has not been raised by the Iranians. But it may be in the future.

Another unresolved issue involves the fate of three more American citizens held in Iran on various charges. One of them is Cynthia Dwyer, a selfdescribed free-lance journalist, who was arrested in Iran last May and charged with spying.

A second American, Mohi Sobhani, a naturalized citizen with business interests in Iran, was arrested last September on unspecified charges.

The third American, also a naturalized citizen, is Zia Nassry, who was dragged out of a hotel room in Tehran 10 months ago and was accused of being a spy for the CIA.

None of the three was among the hostages taken when the American Embassy was seized in November 1979, and none of them was involved in the negotiations leading to this week's hostage agreement. how the American government proposes to deal with the fate of these three Americans is unannounced.

For the time being, the freedom gained by the 52 hostages is principally a matter of national celebration. The National Christmas Tree, unlit the past two holiday seasons because of the hostages, was ablaze with lights last night. The Empire State Building in Manhattan was bathed in red, white and blue lights.

A yellow ribbon four feet wide was stretched 300 feet around the Foshay Tower in Minneapolis with a 40-foot bow and 30-foot streamers. New York City was planning a ticker tape parade. Airlines offered free tickets to the former hostages and their families. A 40-member State Department delegation, led by former secretary of state Cyrus R. Vance, flew to Germany Yesterday to await the Americans' return from Tehran. They are to be joined by ex-president Carter today.

But when the celebrations have ended, the hard problems unresolved with Iran will remain to be faced.