Radical Republicans were the elected officials of the Republican Party during the American Civil War (1861 - 1865) and post-war Reconstruction era. They are most well known for their stance on abolitionism and civil rights legislation, as aimed towards the newly freed slaves of the former Confederacy, which was incredibly radical (hence their name) for mid-19th century America. A number of them also supported other then "radical" causes, such as union rights and women's suffrage, but this doesn't apply to all of them.
For the most part, the era of the Radical Republicans ended with Reconstruction itself. In the following century the Republican party would morph, ironically, into the polar opposite of what they represented when they were known as the Radical Republicans.
Seen through the political lens of the 21st century, the "Radical Republicans" had more in common with the Democratic Party of the present-day United States. Both Radical Republicans and modern Democrats push for broad reforms to the established system, tend to draw support from the North and East, cities, and the well-educated, and see the current balance of power as problematic. Both parties also value(d) equality and modernity over tradition and the establishment, making them the enemies of the entrenched elite.
 Modern usage
The term is also used in modern times to refer to Reagan-era and post-Reagan Republicans - "Goldwater Republicans" - who shifted the party from a generally conservative, i.e. preserving the status quo, position, to one of espousing reactionary politics.