There are several main areas in which American and British spelling are different. The differences are often due to the fact that British English has tended to keep the spelling of words it has absorbed from other languages (e.g., French) while American English has adapted the spelling to reflect the way that the words actually sound when they're spoken.
Words ending in -er/–re
American English words that end in -er often end in -re in British English:
theater or theatre
Note that in American English, theater is the standard spelling, but the -re spelling is also commonly used, especially in proper names (the Lunt-Fontanne Theatre).
Words ending in -or/-our
American English words ending in -or often end in -our in British English:
Words ending in -ize/-ise
Verbs in British English that can be spelled with either -ize or -ise at the end are always spelled with -ize at the end in American English:
apologize or apologise
organize or organise
recognize or recognise
Words ending in -yze/-yse
Verbs in American English that end in -yze are always spelled -yse in British English:
Words ending in a vowel plus l
In British spelling, for verbs ending in a vowel plus l, double the l when adding endings that begin with a vowel. In American English, the l is doubled only when the stress is at the end of the word. Otherwise the l is not doubled:
Words spelled with -oe/-ae
British English words that are spelled with the double vowels ae or oe are usually spelled with just an e in American English:
Note that in American English, certain terms, such as archaeology, keep the ae spelling as standard, although the spelling with just the e (e.g., archeology) is usually acceptable as well.
Nouns ending with -ense/–ence
Several nouns that end with -ense in American English are spelled with -ence in British English:
Nouns ending with -og/–ogue
Some nouns that end with -ogue in British English end with either -og or -ogue in American English:
analog or analogue
catalog or catalogue
dialog or dialogue
The distinctions here are not hard and fast. The spelling analogue is acceptable but not common in American English; catalog has become the US norm, but catalogue is not uncommon; dialogue continues to be preferred over dialog.
Aside from spelling differences, many items and practices have different names in American and British English. To explore further, see British and American and terms.