As 'pig's ear' - Cockney rhyming slang for beer.
As 'in a pig's ear' - an expression of disbelief.
As 'make a pig's ear of ' - make a mess or muddle.
The Cockney rhyming slang version of 'pig's ear' is easiest to explain. It's one of the earliest examples of the form and appears in D. W. Barrett's Life & Work among Navvies, 1880:
"Now, Jack, I'm goin' to get a tiddley wink of pig's ear."
That's easy enough to decipher as "I'm going to get a drink of beer", although you would need a Cockney for an explanation of why 'tiddley wink of pig's ear' was thought to be an improvement on 'drink of beer'. 'Pig's ear' rhymes with 'beer' and that's usually enough for rhyming slang. Franklin's Dictionary of Rhyming Slang lists several alternatives for 'beer' - 'Charlie Freer', 'far and near', 'never fear', 'oh my dear', 'red steer', 'Crimea', and 'fusilier' but 'pig's ear' has always been the most popular.
The version 'in a pig's ear' is also perplexing. It originated in the USA in the 1850s as a variant of 'in a pig's eye'. Both phrases were used as expressions of incredulous disbelief and have the same meaning as 'tell it to the marines'. They may possibly be related to 'pigs might fly'. See this link for more on 'in a pig's ear'.
'Make a pig's ear' is a mid 20th century phrase and means 'completely botch something up; make a complete mess of it'. This is first found in print in a 1950 edition of the Reader's Digest:
"If you make a pig's ear of the first one, you can try the other one."
The expression derives from the old proverb 'you can't make a silk purse out of a sow's ear', which dates from the 16th century. The English clergyman Stephen Gosson published the romantic story Ephemerides in 1579 and in it referred to people who were engaged in a hopeless task:
"Seekinge too make a silke purse of a Sowes eare."
'Make a pig's ear of' alludes to what might be the result if someone did try to make something from a sow's ear - not a silk purse but a complete mess.