Contronyms

A synonym is a word that means the same as another. Necessary and required are synonyms. An antonym is a word that means the opposite of another. Wet and dry are antonyms. While synonyms and antonyms are not in themselves interesting, the complexities and irregularities of the English language sometimes make synonyms and antonyms interesting to explore. Many complexities result from words having multiple definitions. A trivial example is a word with synonyms that aren't synonyms of each other, the word beam, for example, having the synonyms bar and shine. Similarly, some words have antonyms that are neither synonyms nor antonyms of each other but completely unrelated: the word right, for example, having the antonyms wrong and left.

A more interesting paradox occurs with the word groom, which does not really have an antonym in the strictest sense but has an opposite of sorts in the word bride, which can be used as a prefix to create a synonym, bridegroom.

Contronyms are special cases of homographs (two words with the same spelling). Some examples:
anabasis - military advance, military retreat
apology - admission of fault in what you think, say, or do; formal defense of what you think, say, or do
aught - all, nothing
bolt - secure, run away
by - multiplication (e.g., a three by five matrix), division (e.g., dividing eight by four)
chuffed - pleased, annoyed
cleave - separate, adhere
clip - fasten, detach
consult - ask for advice, give advice
copemate - partner, antagonist
custom - usual, special
deceptively smart - smarter than one appears, dumber than one appears
dike - wall, ditch
discursive - proceeding coherently from topic to topic, moving aimlessly from topic to topic
dollop - a large amount, a small amount
dust - add fine particles, remove fine particles
enjoin - prescribe, prohibit
fast - quick, unmoving
first degree - most severe (e.g., murder), least severe (e.g., burn)
fix - restore, castrate
flog - criticize harshly, promote aggressively
garnish - enhance (e.g., food), curtail (e.g., wages)
give out - produce, stop production
grade - incline, level
handicap - advantage, disadvantage
help - assist, prevent (e.g., "I can't help it if...")
left - remaining, departed from
liege - sovereign lord, loyal subject
mean - average, excellent (e.g., "plays a mean game")
off - off, on (e.g., "the alarm went off")
out - visible (e.g., stars), invisible (e.g., lights)
out of - outside, inside (e.g., "work out of one's home")
oversight - error, care
pitted - with the pit in, with the pit removed
put out - extinguish, generate (e.g., something putting out light)
quiddity - essence, trifling point
quite - rather, completely
ravel - tangle, disentangle
rent - buy use of, sell use of
rinky-dink - insignificant, one who frequents RinkWorks
sanction - approve, boycott
sanguine - hopeful, murderous (obsolete synonym for "sanguinary")
screen - show, hide
seed - add seeds (e.g., "to seed a field"), remove seeds (e.g., "to seed a tomato")
skinned - with the skin on, with the skin removed
strike - hit, miss (in baseball)
table - propose (in the United Kingdom), set aside (in the United States)
transparent - invisible, obvious
unbending - rigid, relaxing
variety - one type (e.g., "this variety"), many types (e.g., "a variety")
wear - endure through use, decay through use
weather - withstand, wear away
wind up - end, start up (e.g., a watch)
with - alongside, against

Finding such idiosyncrasies in slang is much easier. The word bad can be used as slang to mean good. The word bomb has two slang meanings: failure (as usually used in the United States) and success (as usually used in the United Kingdom).

Some noteworthy antonyms aren't homographs (words that are spelled the same) but homophones (words that are pronounced the same). Some of these include:
aural, oral - heard, spoken
erupt, irrupt - burst out, burst in
petalless, petalous - lacking petals, having petals
raise, raze - erect, tear down

The word contronym (also antagonym) is used to refer to words that, by some freak of language evolution, are their own antonyms. Both contronym and antagonym are neologisms; however, there is no alternative term that is more established in the English language