Metiche is a popular and very useful slang word in many Spanish-speaking countries for a person who is "nosy" or "meddlesome". It's the perfect word to describe someone who butts into affairs that are not their business.
In the song, Romeo Santos hilariously complains that the problem in his relationship is his girlfriend's mother, calling her a "maldita madre de metiche y bochinchera" (a damn nosy and gossipy mother). This single, punchy word sets the entire comedic and dramatic tone of the song.
Romeo Santos is famous for his romantic bachatas, but this one is a little different. "Suegra" is a hilarious and over-the-top song about a man who has had enough of his mother-in-law. He blames her for all the problems in his relationship, calling her a "maldita madre de metiche y bochinchera" (darn meddling, gossiping mother). He complains that she has a "doctorado en joderme" (doctorate in messing with me) and is a "bendita bruja" (blessed witch) who doesn't want to see him and her daughter happy.
He doesn't just complain, though. He hatches a wild, cartoonishly evil plan. For her birthday, he's going to give her a special gift: "Un buen cafecito, pero envenenado" (A nice little coffee, but poisoned). His fantasy continues with him pushing her car off a cliff so "se la lleva el diablo" (the devil takes her). It's a classic, dark comedy take on the age-old problem of a difficult mother-in-law, all set to a smooth bachata rhythm.