It's not so much the fat content in general you need to worry about, but rather the type of fats present in the food. You need a certain amount of fats as part of a healthy diet. I eat quite a lot of fat really, but I'm very fussy about the types I'll eat. Most people have heard of saturated and polyunsaturated fats frequently mentioned in margarine adverts. The main thing is to steer clear of modified fats... there's something called trans fats (trans unsaturated fatty acids) or hydrogenated/partially-hydrogenated fats which are a serious problem.
Trans fats cause significant and serious lowering of HDL (good) cholesterol and a significant and serious increase in LDL (bad) cholesterol (major clogging of arteries, type 2 diabetes, and other serious health issues).
It' s been said that, whereas 'simple' fats form part of a controlled healthy diet, there is no room whatsoever for the intake of trans fats. Basically, it's a stealth killer.
Anything labelled with "hydrogenated" or "partially hydrogenated" oils/fats should be avoided like the plague.