In politics, the adjective radical means favoring revolutionary changes in the social structure. It’s derived from the Latin radix, meaning root, and it relates to the desire to change society at the roots. The word has historically applied to the political left. While reactionaries (extreme right-wing) seek to preserve or bring back longstanding social structures, radicals (extreme left-wing) seek new structures and revolutionary reforms. There are still progressive and even centrist political parties, notably in Europe, that describe themselves as radical with no connotation of extremism.
But radical is now often used as a synonym of extreme, applicable to both sides of the political spectrum. This is unfortunate because both radical and extreme have useful and distinct meanings in politics, and conflating them erodes the meaning of both. While radical can logically apply to nonleftist positions in favor of uprooting society, phrases like radical conservatism are contradictions.
Outside politics, radical means arising from or going to the root. Again, it should not be confused with extreme, which means (1) of the greatest severity or (2) extending far beyond the norm.
For example, these writers use radical where extreme or even just conservative would make more sense:
[T]he radical agenda of the Republican Party and its vocal Tea Party component brings to mind two widely divergent memories. [NJ.com]
Florida is undergoing a radical conservative transformation under Gov. Rick Scott. [Miami New Times]
The centre-right will not capture any more votes in Scotland until they can present a truly radical new programme. [Scotland on Sunday]
And these writers use radical in its traditional political sense:
The country needs a radical reformer more than ever, but the chances of one emerging are mixed. [Wall Street Journal]
Grassroots activists called at the weekend for a radical shake-up at national level after a disastrous slump in its election fortunes. [Financial Times]
Bin Laden fled to Khartoum in Sudan where it was relatively safe for Muslim radicals and terrorists to operate. [Australian]