Fast-growing People Express Airlines said it will begin daily nonstop service between Washington's National Airport and Newark on Aug. 3 with a one-way fare of $19 evenings and weekends.

The no-frills, Newark-based airline said its standard weekday fare will be $44 each way. The $19 one-way fare is available after 7 p.m. weekdays and on weekends. The airline will operate five flights in each direction each day.

People Express's major competitor on the route--New York Air--now charges $59 each way during the week and $39 on weekends until 5 p.m. Sunday. Eastern Airlines' fare is $60 during the week.

The start-up of People Express flights at National was made possible by its purchase of five landing slots from Altair Airlines last month through a government program that allowed airlines six weeks to buy and sell slots at major airports. People Express spent nearly $1.8 million for the National slots, and another $240,000 for 16 slots at Newark and three at Boston.

The airline, which was started by five former officials of Texas International Airlines, got off the ground late in April 1981 with three planes serving four cities. Its 17 Boeing 737 aircraft now serve 12 cities. A profitable airline already, People Express last week announced a new offering of 1.5 million shares of common stock designed to raise between $20 million and $30 million to buy new aircraft.

In an effort to keep its costs low, the airline has eliminated ticket counters, airline food and "free" baggage check-in. Passengers book reservations either directly or through travel agents and proceed to the gate for check-in. Tickets can be purchased on board the aircraft. Travelers are encouraged to carry their luggage and place it in oversized overhead compartments, but can pay $3 per bag extra to check it.

Because of the airline's low cost structure, its new flights on the 118-passenger 737s will be profitable with either 100 passengers at the $19 fare or 50 people at the $44 fare, People Express President Donald C. Burr said yesterday.

People's entry on the Newark-Washington route against New York Air means that two of the airlines formed after passage of the Airline Deregulation Act of 1978 will be pitted against each other for the first time, and it promises to be a vigorous competition.

"We're the incumbent here on the route; it's a very interesting situation," New York Air President Michael E. Levine said yesterday. Washington-Newark "is the only market in which we can say we are the dominant carrier," he noted.

"Our view is basically that People Express is a very different airline from us" and that the two may establish very different clientele, he said. NYA aims special services at business travelers, providing increased leg room and free beverage and snack service.

Levine said he wasn't knocking People's service. "It's a perfectly respectable K mart product," he said.

He added that NYA was announcing today a new $39 weekday fare on the route that is available if purchased seven days before departure; the fare was "already in the works," he said. NYA, which isn't profitable yet, has five round-trip flights a day on the route now, and will have six beginning Aug. 1.