Trans World Airlines' parent company signed a definitive agreement yesterday to acquire for $88 million the Century 21 Real Estate Corp., which has grown during just seven years of business to become the world's largest real estate sales organization.

Directors of both firms approved the takeover plan yesterday, under which Century 21 would become a wholly owned subsidiary of Trans World Corp., which also owns the Hilton hotel chain, Canteen Corp. and Spartan Food Systems.

In an unrelated announcement, May Department Stores Co. of St. Louis announced approval of an agreement to buy Volume Shoe Corp. of Topeka, the largest self-service U.S. retail shoe chain with 762 stores in 32 states. May, owner of the Hecht Co. here, is the nation's second largest department store retailer.

Trans World said it will offer several options to current Century 21 stockholders in its acquisition.The Century 21 owners can take $28.50 in cash for each share they own; exchange stock for Trans World convertible preferred stock with an equivalent value of $28.50 a share; or receive a combination of 40 percent in stock and 60 percent in cash.

Overall, Trans World said, not more than 49 percent of the Century 21 shares may be exchanged for cash. As of last Friday, there were 3.1 million shares of Century 21 outstanding; the company first sold public shares in 1977.

After the merger is completed, Century 21 Chairman Arthur Bartlett would become a Trans World director.

Based in Irvine, Calif., Century 21 has prompted a revolution in the American real estate sales business since 17 offices were opened initially in Orange County, Calif., in 1972.

Today, there are 7,500 Century 21 brokerage offices-mostly franchises employing 75,000 agents in this country and Canada. For the 12 months ended June 30, Century 21 earned more than $5 million on revenues of $36.5 million.

Of all previously-occupied houses sold in the country each year, Century 21 is the agent for more than 8 percent-one of every dozen houses.

Until the concept of franchised local real estate businesses was developed by Century 21, based on the success of national fast-food chains, small enterpreneurs dominated real estate sales in local communities.

Bartlett, himself a veteran real estate salesman, was behind the idea that management training, advertising and marketing techniques could be used to replace the traditional small, local broker with a familiar national image. In exchange for its expertise, Century 21 collects an initial franchise fee (which was $7,000 in 1978 from each member broker) plus a service fee of 6 percent of annual gross sales.

The annual service fees are applied to Century 21 regional marketing. In addition, franchise owners contribute to the multi-million dollar advertising campaigns that have made Century 21 a real estate name from coast to coast.

About a million sales people and 665,000 brokers are engaged in real estate sales throughout North America. Real Estate offices are becoming larger and ownership is expected to become substantially concentrated.

Trans World, which reported profits last year of $87 million on revenues of $3.7 billion, said stockholders of both firms will meet Oct. 18 to vote on the proposed combination.

The May Co. agreement calls for an exchange of 1.35 shares of May common for each share of Volume Shoe, which has announced plans to open 150 new stores in the next year. Stockholders will vote on the merger plan Nov. 16.