Mad has two everyday meanings and they belong to different countries. In Britain it usually means insane or wildly foolish — a mad idea, gone mad, mad about a hobby. In America it usually means angry: don't be mad at me.
That split is the single most useful thing to know about the word, because British mad at somebody sounds like a mistake and American mad meaning insane sounds old-fashioned. Both dialects share mad about, meaning enthusiastic: mad about horses. On register: mad for insanity is informal and now often avoided when talking about actual mental illness, where the polite words are different. As an intensifier it is common and harmless — mad keen, driving like mad.
1adjectivemadIn American use, angry.
“She was mad at him for a week.”
2adjectivefoolishFoolish, or wildly unwise.
“Driving in that weather was mad.”
3adjectiveinsaneMentally ill — an old and now avoided use.
“In the story he goes mad in the last act.”
synonyms: insane
4adjectiveextremelyExtremely enthusiastic about something.
“He is mad about trains.”
Too few real examples of this word to estimate which sense is commonest.
mad is a state of mind.
Also accepted in games: angry, crazy
Old English gemaed came from a Germanic verb meaning to damage or cripple, so the oldest sense is of a mind that has been injured rather than one that was always so. That is a bleaker picture than the modern casual use, and it explains why the word still lands harder than it looks.
Middle English
mad
English, today
mad
Built from
From Middle English mad. From Middle English mad, madde, madd, medd, from Old English ġemǣd, ġemǣded (“enraged”), past participle of ġemǣdan, mǣdan (“to make insane or foolish”), from Proto-Germanic maidijaną (“to change; damage; cripple; injure; make mad”), from Proto-Germanic maidaz "weak; crippled"; compare Old English gemād (“silly, mad”), Old High German gimeit (“foolish, crazy”), literary German gemeit (“mad, insane”), Gothic 𐌲𐌰𐌼𐌰𐌹𐌸𐍃 (“crippled”), from Proto-Indo-European mey- "to change"; compare Old Irish máel (“bald, dull”), olt ap-maitinti (“to wound”), Sanskrit मेथति (“he hurts, comes to blows”).
Etymology from Wiktionary, CC BY-SA