The word radical comes from the Latin word for "root." This gives you some idea of what it means in the political sense. Radicals generally seek root-and-branch change of a political system, getting rid of the old and replacing it with something brand new, completely different, and much better.
A good example of what radicalism means in practice is given to us by the French Revolution of 1789. The French Revolutionaries were radicals in that they got rid of a political system that had existed for centuries and replaced it with a completely different one based on the abstract philosophical principles of liberty, equality, and fraternity. Whether or not the new system was a good idea, it certainly represented a big change over what preceded it.
In contrast to radicals, reactionaries do not favor change. Of course, they recognize that some changes are unavoidable, even sometimes necessary. But they seek to minimize change as much as humanly possible.
Reactionaries tend to look at societies and political systems as complex organic structures that need to be allowed to develop slowly and naturally over time. They differ from radicals, who want to rip up everything by its roots and start again from scratch on the basis of abstract concepts. Reactionaries regard this as a dangerous notion that leads to instability and chaos.
The violent phase of the French Revolution known as the Terror has often been cited by reactionaries in support of their arguments. Despite its grand, noble rhetoric, the Revolution degenerated into violence, bloodshed, and chaos, with many innocent people being sent to the guillotine.
To reactionaries, this is what happens when you disturb the delicate balance of society on which order, stability, and respect for the rule of law are based.