Basics of why PUFA is bad:
Saturated fats are molecularly stable; resistant to degradation. PUFAs, by nature of their molecular arrangement, are less stable and more reactive to things like oxygen. A PUFA oil left open on the countertop will quickly react with the air and quickly go rancid, while a saturated fat would remain fresh and intact.
http://img.photobucket.com/albums/v53/Flare8/he/satvsunsat.png (don't take this second picture to imply monounsaturated fats are unsafe, just try to get the image in your mind of how PUFAs may be inherently unstable).
The body is a hot and busy place where stable fats can run safe but unstable fats are prone to breaking apart. When PUFAs do break apart in your body, its like glass shattering and all the shards damage whatever they come in contact with: damaging blood vessels, interfering with signal receptors, interrupting cell's energy production, etc.
One of the reasons PUFA damage seems long-term rather than immediate may be because PUFA stacks up in your fat tissue and accordingly does progressively more damage. Your body can deal with some degradation of fats (e.g. oxidation), but too much combined with too little nutrition (e.g. weak antioxidant powers) means you become overloaded and start breaking down as an organism.
PUFAs role in nature:
When you cool saturated fats (like butter), they become solid. Plants, grains, and certain cold-water fish are often in cold environments, so if they had saturated fats they would harden up like a chilled stick of butter and that would be bad for their survival. They need some fats so PUFA is their only choice. Note how plants that live in warm climates, like coconuts, carry predominately saturated fats. Humans are warm-blooded so we too are primed for saturated fats.
Some animals, called 'ruminant' animals such as cow or lamb, have a lot of evolutionary experience consuming grasses, grains, etc. They somehow have mechanisms to detoxify or otherwise deal with PUFAs and convert them into saturated fats. You can feed a cow grains and its fat will not be filled with PUFA.
Other animals (non-ruminants) such as humans or chickens, do not have evolutionary experience consuming grains, and they do not have mechanisms for detoxifying/dealing with much PUFA. Feeding them PUFA causes PUFA to stack up in their fat tissue.
The human body may use small amounts of certain PUFAs for some signaling purposes. An excess of certain PUFAs (e.g. omega 6) may translate into an excess of certain signals, which may cause bodily dysfunction.
I think all PUFA consumption should be limited. Some say omega 3 PUFA supplementation is desirable in order to balance the grain-derived excessive omega 6 PUFA, but here are studies i've come across suggesting harm from omega 3/fish oil http://flare8.net/health/doku.php/misc#fish_oil . Clinical trials supplementing omega 3 sometimes see good benefits, but I think the same benefits could be accomplished by reducing omega 6 (i.e. corn oil) consumption. I think the safe 'do what's evolutionarily accordant' thing is to seek overall low intakes of PUFA.
Ruminant animal fat is always acceptable. Non ruminant animal fat is unacceptable if the animal was fed corn soy etc, but is acceptable if they were fed natural diets (like a 'pastured' chicken). Corn/soy feeding is the norm in USA. Sometimes its different in other places.
I would buy supermarket beef but would make an effort to buy pastured eggs. If I had to shop at a supermarket I would prefer beef and lamb over chicken and pork fats. I would eat skinless chicken because that has almost no fat in it.
Here are some PUFA levels for various foods. Note that 'chicken' is high in PUFA, but this is only true if the animal is grain-fed:
A good, more technical article on PUFAs: http://www.westonaprice.org/know-your-fats/precious-yet-perilous
Various scientific references against PUFAs: http://flare8.net/health/doku.php/fats#polyunsaturated_fats_pufa