What is the name of the first American magazine?

The first American magazine was called... The American Magazine. It was published in Philadelphia, beginning on February 13, 1741. It wrote about the colonial government, and also covered religion, current events, and some history. Andrew Bedford was the publisher. Benjamin Franklin, who had hoped to get there first, ended up publishing the second magazine in America, the General Magazine, on February 16, 1741; as it turned out, Franklin's magazine lasted longer than Bedford's did.

I am not sure what you mean by "significant," but magazines really expanded in the mid 1800s, as transportation improved (such as the railroad, to carry magazines from city to city in a more timely fashion); and the telegraph made it possible to report and then publish breaking news much more quickly. Magazines like the Atlantic Monthly (1857) and Harpers Bazaar (1867) became extremely popular with large audiences, and other magazines soon followed.