There are multiple spelling differences between American and British English, and speakers of both often end up misusing certain words and spellings. This article covers some of the more common spelling differences, it is not an exhaustive list but should give you a good idea of how to deal with other, similar words.
In the following "vs" examples, the US spelling comes first.
Color vs Colour (-or vs -our)
An extra u in British spellings is one of the most common spelling differences. Color vs colour is a common one to see around the internet though you'll also come across this spelling difference in words like flavor vs flavour, humor vs humour, honor vs honour and so on with the American spelling leaving out the u.
A notable exception to this rule is glamour. While glamor is acceptable in US English, for some reason the British spelling seems to be used more often.
Gray vs Grey
This is one of the spelling differences that confuses people on a regular basis, whichever version of English they speak. Generally speaking, gray is the American spelling and grey is the British spelling but you'll see plenty of British people using gray and vice versa. Grey tends to be the spelling of choice in Canada.
Licence vs License
In US English, license is used both as a noun and a verb. In UK English, licence is only the noun and license is used as the verb.
American: Clive hoped the licensed instructor would soon award him his license.
British: Clive hoped the licensed instructor would soon award him his licence.
Meter vs Metre (-er vs -re)
Once again, this is one of the spelling differences you'll see on a regular basis. Center vs centre, theater vs theatre, fiber vs fibre etc.
Unfortunately, this spelling difference isn't something you can completely rely on since many words in British English actually use the -er ending. See letter, number and proper as examples of this awkwardness.
Organize vs Organise (-ze vs -se)
Another common spelling difference between US and UK English is the use of -ze over -se. In UK English, -se is the accepted usage as in organise, recognise, serialise etc. In US English it's -ze, as in organize, recognize, serialize and so on.
Practice vs Practise (-ce vs -se)
The spelling difference between practice and practise is similar to that between license and licence in that UK English uses a different spelling for noun vs verb, whereas US English uses the same spelling for both.
In this case, apparently just to confuse us, the American spelling uses practice (note the -ce ending) for both whereas with license the American spelling uses the -se ending. The British usage is practice for the noun and practise for the verb.
American: Clive practiced hard, and hoped one day to open his own practice.
British: Clive practised hard, and hoped one day to open his own practice.
Other common spelling differences