American English and British English are the two main dialects (types) of the English language. American English is spoken in the United States, while British English is spoken in the United Kingdom. Both types of English are different to each other in some ways, for example in spelling, punctuation, grammar and vocabulary.

Spelling 

There are many words that sound the same in both American and British English but have different spellings. British English often keeps more traditional ways of spelling words than American English. For example:

Most words that end in -our in British English (colour, labour) end in -or in American English (color, labor).
Most words that end in -re in British English (metre, centre) end in -er in American English (meter, center).
Most words from Greek and Latin that are written with ae or oe in British English (anaemia, diarrhoea) are written with an e in American English (anemia, diarrhea). But the British spelling archaeology is still common in American English, while the American spellings encyclopedia (from Latin encyclopaedia) and medieval (from Latin mediaevalis) are common in British English.
Verbs that may end in -ise or -yse in British English (analyse, realise) always end in -ize and -yze in American English (analyze, realize). The Oxford English Dictionary (a dictionary for British English) use verbs end in -ize and -yse (analyse, realize).
Nouns that end in -isation in British English (organisation, realisation) end in -ization in American English (organization, realization). The Oxford English Dictionary use nouns end in -ization.
Most words of French derivation which end in -ogue in British English (analogue, dialogue) end in -og in American English (analog, dialog). However, some words in British spelling, for example dialogue, demagogue, are still common in American English.
Inflected words that are written with double l in British English (cancelled, travelled) are written with a a single l in American English (canceled, traveled). This spelling change is not always followed in American English (cancellation).
Some inflected words written with a silent e at the middle in British English (ageing, judgement) are written without the silent e in American English (aging, judgment). This spelling change is not always followed in American English (changeable).
Some past tense verbs end in -t in British English (learnt, spelt) end in -ed in American English (learned, spelled). The final letter is usually pronounced in both cases.

Because some chemical elements have different spellings in American and British English, IUPAC has to use only one spelling of a word in their papers. For example:
Aluminium in British English becomes aluminum in American English. IUPAC uses the British spelling aluminium.
Caesium in British English becomes cesium in American English. IUPAC uses the British spelling caesium.
Sulphur, sulphate and sulphide in British English becomes sulfur, sulfate and sulfide in American English. IUPAC uses the American spelling sulfur.

Punctuation
Punctuation is the name for the marks used in writing text. Punctuations, for example brackets/parentheses, full stops/periods, quotation marks/inverted commas, have different rules in American and British English.

Brackets/parentheses
In British English, "( )" marks are sometimes called "brackets", "[ ]" are called "square brackets" and "{ }" are called "curly brackets". In formal British English and in American English, "( )" marks are "parentheses" (singular: parenthesis), "[ ]" are called "brackets" or "square brackets", and "{ }" can be called "curly brackets" or "braces". These marks are used in the same way in both dialects.

Quotation
The rules on quoting a text is different in American and British English. In American English, " and ' are called "quotation marks". In British English, " and ' are called "inverted commas" or "speech marks". In American English, it is common to use the double quotation mark ( " ) in direct speech. In British English, it is common to use the single inverted comma ( ' ).

Related articles 
 List of dialects of English
 Oxford spelling

References

Other websites
 Word substitution list, by the Ubuntu English (United Kingdom) Translators team
 Linguistics Issues List of American, Canadian and British spelling differences
 Map of US English dialects
 The Septic's Companion: A British Slang Dictionary
 
 British English vs. American English Slang Compared
 British English-American English Vocabulary Quiz

American English
British English