Ibuprofen (Advil, Nuprin etc) and Naproxen (Aleve) are both NSAID's like aspirin. They therefore share many of the potential side effects that have already been highlighted as been endemic to aspirin. For instance, aspirin allergy, upset stomach and a higher risk of asthma attacks can be equally ascribed to both of these medicines as they can to aspirin.
Ibuprofen however carries additional risks as it is believed to increase the risk of myocardial infarction (heart attack) if taken in high dosages and you can also cause or exacerbate irritable bowel syndrome, Crohn's disease and ulcerative colitis due to its capability of starting gastrointestinal bleeding.
With Naproxen, some of the most widely reported side-effects include constipation or diarrhea, drowsiness, upset stomach, stuffy nose and heartburn.
However, whilst the NSAID's already detailed are likely to carry a slight increase in risk levels for heart attacks and strokes, the risk associated with Naproxen might be considerably more serious. Indeed, the National Institute of Health terminated a trial into the effects of Naproxen as a drug for reducing the severity of Alzheimer's disease (in 2006) because of the suspicion that the drug significantly increased the risk of heart attack or stroke.
Since that time, further analysis suggests that at normal dosages, the increased risk of suffering a heart attack or stroke is fairly low although as you will see in the next section, this is certainly not true of all NSAID's.