A. Technological changes - places where the Halacha is based on what we today perceive as faulty science. The example is "Metziza Ba'peh" - the sucking of the blood performed orally after circumcision, which originally was believed to be healthy, and now is considered a major source of infection.
B. Lifestyle changes - When a Halacha is based on assumptions on the social norm. The example given is the time for saying Shema - which was defined as "when people wake up" and "when they go to sleep". Today very few people's day-cycle is closely linked to sunrise and sunset, and hence the halachic definitions are out of sync with modern life.
C. Ethical changes - Halachot that reflect ethics - the example the article gives is beating your wife, and giving a Get (Divorce). You can also think of various halachot of Tzniut such as head coverings, which there is a strong reason to believe were reflective of the culture of the time, and not a "Torah" ethic.
I think there is a fourth case which many people mean when they speak of modernizing halacha, but which in reality is a combination of A+B:
D. Technological Lifestyle Changes - The obvious example is electricity on Shabbat. No matter how many times you explain the rational for electricity being forbidden on Shabbat, people are not really convinced. To me it seems somewhat obvious that when the Chazon Ish decided that electricity on Shabbat was forbidden he was thinking of certain acts - namely working, Cars etc. which he was determined not to allow on Shabbat. However, in today's reality light bulbs are not seen as any type of work, or interfering with a day of rest (though some televisions can be considered a pumped up light bulb, and still be considered not suitable for Shabbat).
It is important when you are having the argument about Modernizing Halacha that you know which argument you are having. All too often the two people arguing are not aware that they are arguing different arguments. Additionally, there is no logical reason that you have to give the same answer to every one of the four meanings of the term. You can believe that halacha should be based on modern science, but that the ethics that halacha espouses are eternal truths. Or you could believe that Chazal had a perfect knowledge of the physical world, yet some of their halacha reflected their social norms. Any of these combinations are legitimate, but they all lead to different answers on what halachot you believe should "modernize".