Primary Differences in Transatlantic Spelling

Generally speaking, it's pretty easy to figure out the British spelling of any American word or vice-versa (usage differences are beyond the scope of this writeup). There are a few primary differences, outlined below.

Verbs

"-Ize" vs. "-ise."
In American English, only "-ize" is used (with a couple of oddball exceptions like "improvise" and "surprise" {Yeah, I know, it's not really in the same class, but it appears that way nonetheless}). In British English, "-ise" is preferred, but "-ize" is acceptable.
< e.g. Am "realize" Br "realise" or "realize"

Irregular Past Tense
Several English words can either be spelled with the normal "-ed" ending or with a "T" instead (e.g. "learned"/"learnt," "burned"/"burnt"). American English tends to prefer the "-ed" endings while Britons are more fond of "T" endings, although they are interchangeable in either dialect.

Nouns

"-Or" vs "-our" nouns
The canonical example of the difference between British and American English. For many words, Americans would use "-or" and Britons "-our." Of course, this isn't always true, so don't go dropping the "U" from "hour." Another important distinction: In Britsh English, the "O" is dropped when the word is made into a verb, e.g. "colorise" and not *"colourise."

Greek words
There are a number of instances where Americans will drop the "O" in the "oe" from a Greek word in which it makes the "-ey" sound in "key" while British English will retain it (e.g. "fetus" versus "foetus") but it seems the pioneers of American English kind of did this willy-nilly, since the word "onomatopoeia" keeps it and "amoeba" in American English is more common than "ameba," though the latter is considered acceptable. Above all, rely on aesthetics.

"-Gue" vs "-g"
"-Gue" endings for words like "dialogue" are preferred in both American and British dialects, but only American English allows "dialog." Same deal with "catalogue" and "catalog" and some others, I'm sure.

"-Re" vs. "-er"
British spelling only uses "-re" in words like "kilometre"/"kilometer", American, only "-er" (except for "center"/"centre" and "theater"/"theatre"), where both are acceptable). However, for a word like "killer" this is simply false. If you can imagine someone saying "-a" instead of "-er," then in British English it probably uses "-er."

"Ls" Sometimes British words and American spellings will use different numbers of "Ls," like "traveller"/"traveler." Britons tend to double up on the "Ls."

Other weird noun stuff
"Moslem" is more common in British usage (Fisher says that this is old-fashioned and not so true anymore), but "Muslim" is more common in American usage. Both are acceptable.
Am "maneuver" as opposed to Br "manoeuvre"
When transliterating Arabic, Americans are more likely to use "I" for the "-ey" sound in "key" (e.g. Mazar-I-Sharif) whereas the Brtions prefer an "E" (e.g. Mazar-E-Sharif). However, either is acceptable.
Am "tire" is equivalent to Br "tyre"

Adverbs

"Ls"
With some adjectives and adverbs, Americans use double "Ls" and Britons don't necessarily.
e.g. Am "skillful" vs. Br"skilful" or "skillful"

Abbreviations
Americans use periods after abbreviations like "Mr." Britons don't necessarily.

I'm an ignorant American, so /msg me with corrections or additions, you crazy Redcoats.