In the United States, food picked up at a restaurant and eaten elsewhere is takeout. In the U.K., the word is takeaway. Neither is right or wrong. It is simply a dialectal difference.
Canadian writers favor takeout, though takeaway appears occasionally in Canadian publications. Australians and New Zealanders use takeaway.
Both takeaway and takeout are one word when functioning as a noun (e.g., let’s get some takeout) or an adjective (e.g., let’s get some takeaway sushi). As verbs, the phrases take out and take away are two words.
Takeaway is sometimes pluralized—takeaways—to refer to multiple takeaway meals or establishments that sell takeaway food. Takeout is always singular; it’s a mass noun.
British, Irish, Australian, and New Zealand publications tend to use takeaway—for example:
Those who love Chinese takeaway will be pleased to hear you can grow delicious Oriental vegetables at home. [Daily Mail]
Built around Vietnamese, Thai and Laotian snacks and salads, it’s effectively a takeaway product with a cult foodie following. [The Australian]
For example, we used to both ritually buy at least one takeaway coffee a day. [Stuff.co.nz]
Americans and (to a lesser extent) Canadians prefer takeout, as used in these sentences:
Although it appears a takeout meal for $4.99 is a great deal, one would actually save money by going to the store. [USA Today]
At Mambella’s, a popular Italian takeout restaurant in the heart of RIM’s Waterloo campus, it was business as usual. [Hamilton Spectator]
Takeout shops and restaurants around the city will gladly pack a lunch or supper to enjoy in the park. [New York Times]