This bit of British slang refers to an argument, a row, a disputatious bandying of words back and forth between two people.
Argle-bargle is widely used in the UK, as its many variants attest: argy-bargy, argie-bargie, argue-bargue, or simply argy, or bargy. These forms suggest a diminished importance of the arguing referred to, which is to say, an argument that shouldn't be taken seriously, a spat. However, it belongs to the realm of slang and hence should not be used in your college applications.
The thrust of argle-bargle is a mild argument, even a civil debate: "Archie should have been a lawyer rather than a salesman; he enjoys the argle-bargle of contract negotiation more than closing the deal." Since this funny word is colloquial rather than formal, you can push the envelop of English grammar a bit when you use it. Mom might warn the kids, "OK, kids, cut the argy-bargy before it becomes pushy-hitty."
This funny word is a nonsense rhyming compound (like willy-nilly, piggly-wiggly, boogie-woogie) made up of two rhyming words. Their history is interesting: argle emerged in the 16th century as a blend of argue and haggle. Bargle was added much later, in the early 19th century, simply because it rhymes with argle. The spelling has varied between what you see here and a slightly Greekier argol-bargol, but the spelling given here is now the most widely used.