Both acknowledgment and acknowledgement appear throughout the English-speaking world, but acknowledgment, without the middle e, is preferred in U.S. and Canadian English, while acknowledgement is preferred outside North America. These preferences extend to the plural forms, acknowledgements and acknowledgments.
In the U.S. and Canada, acknowledgement appears about once for every two instances of acknowledgment. The ratio is the other way around in British and Australian publications, at least the ones that make their content searchable online. So wherever you are writing, both forms are common enough to be considered acceptable, but it may be safer to stick with the one that your readers are more likely to consider correct.
Acknowledgment is now considered the American spelling, but it was preferred in all varieties of English until recently. It’s not necessarily older, as instances of both spellings are easily found in texts going back to the 16th century when the word entered the language. But as shown in this ngram, which graphs the use of both forms in British books, magazines, and journals published from 1800 to 2000, British writers once favored acknowledgment by a wide margin, and that form gave way to the longer spelling toward the end of the 20th century:
Though both forms can be found everywhere, acknowledgment is preferred in American and Canadian publications such as these:
The acknowledgment of a possible al Qaeda role came at a congressional hearing in which government officials were peppered with questions. [Wall Street Journal]
I yearned for his attention and acknowledgment. [Globe and Mail]
It may seem odd for the author of a book on human genetics and heredity to thank his travel agent in the acknowledgments. [New York Times]
Outside North America, acknowledgement is preferred—for example: