The spelling of some words varies among different English speaking groups. Just like color and colour, favourite and favorite, Both acknowledgement and acknowledgment are correct. So, both can be considered as the correct spelling.
In both British English and American English, many words having the same meaning may have different spelling. However most of the time one variant is preferred over the other.
The correctness depends on which geographical location you are in and which English you are using. Acknowledgement is used in British English and acknowledgment is used in American English. In North America, it is spelled without an ‘e’ in the middle, whereas in the UK, there is an ‘e’ in the middle.
Both the forms can be considered correct and acceptable wherever you are writing. However, using both versions of the spelling in a single article or a single paper should be avoided. For consistency, you should consider using only one variant of the spelling while writing.
Similar to Acknowledgement and Acknowledgment, there are a lot of other words whose spelling has a different version in British and American English. Some of the words mentioned in Wikipedia having variation are:
- acknowledgement vs acknowledgment: acknowledgement is preferred in British English, acknowledgment in American English.
- judgement vs judgment: judgement is preferred in British English (except in the sense of a judge’s decision, in which case judgment is preferred), judgment in American English.
- per cent vs percent: per cent is preferred in British English, percent in American English.
- dialogue vs dialog: In a non-technical context, the spelling dialogue is preferred in American English. In Webster’s dictionary, dialogue is given first, and Chambers also indicates dialog is less used in North America.
- catalogue vs catalog: Webster’s treats this case differently, as does Chambers —catalog is the preferred spelling in American English.
- glamour vs glamor: The spelling glamour is preferred in both British and American English. (Glamourous is sometimes found in American English, but is usually considered incorrect in British English, where glamorous is the only accepted form.)
- foetus vs fetus: In American English, foetus is never used. In British English, usage is divided. In academic literature, fetus is preferred.
Similar to these variations in spellings in the words having the same meaning, acknowledgement, and acknowledgment both the spelling are correct. As mentioned above, you can use both spellings but only use one variation for consistency.