The nation's civilian unemployment rate dropped slightly in April to a seasonally adjusted 10.2 percent as strong gains in employment more than offset an increase in the number of people actively seeking work, the Labor Department reported yesterday.

The employment report indicates that the nation's economic recovery broadened and moved ahead in April, with payroll employment increasing by 260,000 jobs during the month. Since December, when unemployment was 10.8 percent, payroll employment has increased by 650,000. The unemployment rate in March was 10.3 percent.

President Reagan, addressing an audience of elderly crime fighters in Sun City, Ariz., said he was baffled by the April unemployment numbers. "Now, I have to confess to you that there are things about the unemployment statistics that still baffle me because there's no change in the unemployment percentage rate," he told the group. "But there are 250,000 more people working in April than were working the month before. And since December, there have been 650,000 added to the list of those employed."

If members of the armed services are included, the unemployment rate was 10.1 percent for both March and April. The rate was unchanged because many discouraged people who had not been looking for work began to do so, a common occurrence in recoveries when jobs become more readily available. More people found jobs last month, but because more people also were looking for jobs, the unemployment rate stayed essentially the same.

More jobs were available. The Labor Department said three-fourths of 186 industries surveyed for the April report said they hired more workers last month, and the length of the average workweek rose as well. The total number of hours worked in April rose about 0.8 percent.

Despite the employment gains last month, 11.33 million people looking for work were unable to find it, a decline of only 53,000 from March. Unemployment among blacks rose nearly a full percentage point to 20.8 percent, tying the record levels of last fall. The rate for whites dropped from 9 percent to 8.9 percent.

Economic forecasters are confident now that the recovery is on solid ground but that it is likely to continue at a pace lower than that following most previous postwar recessions.

The forecasters generally believe that the gross national product will rise at about a 4 percent to 5 percent annual rate this quarter, up from a 3.1 percent rate in the first three months of the year. The April figures on employment and hours worked are consistent with those predictions, analysts said.

Increases in the output of goods and services in the second half of 1983 could be larger than in the first half if consumers step up their spending as expected, they said.

The housing sector, which has been the strongest part of the economy lately, could get a boost from a cut in mortgage interest rates with FHA and VA insurance announced yesterday by the Department of Housing and Urban Development. HUD said that, effective Monday, the FHA and VA mortgage interest ceilings would be dropped half a percentage point to 11 1/2 percent.

"Factory jobs rose by 110,000 from March to April," Janet Norwood, commissioner of labor statistics, told the Joint Economic Committee. "Most of the increase occurred in the durable manufacturing industries, which had been hard hit during the recession. In the four months since December, sizable job gains have occurred in lumber and wood products, primary and fabricated metals, electrical equipment, and transportation equipment."

Meanwhile, employment in service-producing industries, which changed little during the recession, has begun to grow again. Since December, employment in that sector is up by 450,000, Norwood said, with gains of about 100,000 in both March and April. Construction employment was also up by about 30,000 last month, while the number of jobs in mining, retail trade and government all fell somewhat.

Most of the increase in the size of the labor force came among adult men, who had stopped looking for jobs in large numbers back in January. As a result, the unemployment rate for adult men rose from 9.6 percent to 9.8 percent last month, despite the increase in employment in manufacturing jobs, which are held predominantly by men. The rate for adult women, on the other hand, fell from 8.8 percent to 8.4 percent as the increases in their employment outstripped the rise in the number of women in the labor force.