I recently started proofreading for a local magazine and frequently come across the case whereby the word 'and' is directly followed by a comma. Is this correct punctuation or are there specific cases when this should be used- i do not recall ever learning this at school?
No, "and" shouldn't be followed by a comma, presuming you mean as a word in a list:
"The weather lately has been cool, unsettled and, stormy."
This is a hopelessly wrong use of the comma.
The right punctuation is either one of these (depending on which Theory of Commas you go with):
Serial comma - Wikipedia, the free encyclopedia
The weather lately has been cool, unsettled and stormy.
The weather lately has been cool, unsettled, and stormy.
In the case where a pair of commas are setting off a phrase, it could occur that the phrase begins after the word "and." That would be all right.
"The weather lately has been unsettled and, as the entire world knows, damaged by global warming.