Trustees of American University yesterday approved an 18 percent tuition increase for next fall, the latest in a series of major increases announced by colleges in the Washington area and around the country.

The increase, which will raise tuition and required fees at AU to $6,270, was approved despite a protest by about 2,500 students two weeks ago.

The rise in tuition rates is occurring at the same time the Reagan administration has proposed substantial cuts in federal student aid. As a result of the two developments, a campus movement has begun to emerge.

Yesterday about 5,000 students from across the country, including a delegation from American, took part in a day of lobbying and rallying on Capitol Hill to oppose administration proposals to cut student aid.

At George Washington University, where tuition is going up 19.5 percent to $4,900, a one-day class boycott was organized by the student government. But officials said virtually all classes and midterm exams went on as scheduled.

About 500 George Washington students marched to the late afternoon Capitol Hill rally, carrying an effigy of President Reagan and banners reading, "Books Not Bombs." Speakers at the rally, who pledged strong support for student aid, included House Speaker Thomas P. O'Neill, Rep. Peter Peyser (D-N.Y.), a leading organizer of the protest, and Sen. Alphonse D'Amato (R-N.Y.).

At the AU trustees meeting, student confederation president Don McEachin said the tuition increase along with the aid cuts might force some students to drop out.

According to a spokesman, university president Richard Berendzen said he was "very sympathetic" to the students' problems, but that the increase was necessary to enhance university programs. The university is planning to raise its own student aid proportionately with the tuition increase and add $200,000 more, said spokesman Jody Goulden.

Even so, she said, university officials expect an enrollment drop of about 75 students from the current 8,695 full-time equivalent enrollment, but she said most of this decrease would result from tougher admissions standards, not from students' financial problems.

With increased costs for room, board, and transportation, Goulden said she expects the total price tag for a year at American University to reach about $10,500 next fall, about $5,000 more than at most state colleges, including the University of Virginia and the University of Maryland.

Among other local private schools, Trinity College in Northeast Washington recently announced an 18 percent tuition increase, while a campus finance committee has recommended a similar increase at Georgetown University where trustees are scheduled to take official action in mid-March.

Around the country, Harvard University, traditionally one of the nation's most expensive, recently announced a 14.8 percent tuition increase, bringing its total estimated cost for a year to about $13,700. Although no national survey is yet available, officials of the College Board said university price increases for next fall appear to be running ahead of this year when they averaged a record 13 to 14 percent.

Lawrence E. Gladieux, executive director of the College Board Washington office, said the increases may be a "catch up" for relatively modest rises below inflation rates in the late 1970s. He said the massive increase in low-interest federally guaranteed student loans, which have more than tripled since 1978 to $7.7 billion a year,"may have paved the way" for the boosts.