This is a great question. However, it is difficult to define radicalism and for this reason radicalism can take many different shapes. But what radicalism has in common is that it does not seek to go about things in the traditional or accepted way by society. This is why it is called radical. Perhaps some examples would be helpful.
Martin Luther King's ideas of peaceful demonstrations or his non-violent approach to social change and the civil rights movement are two examples of radicalism in the modern world. These were new ways to seek change in society as social changing was not taking place fast enough.
A more recent example would be terrorism. People talk about radical Muslims and their approach to social and economic change. They believe in targeting even civilians to make their point. The attacks on 9/11 were such attacks and the ideology was rooted in radical Islam.
In light of these two example, we can say that there is a broad sweep in radicalism. Hence, each one has to examined on a case by case basis.