Roads and railroads allow units to move longer distances within a turn by reducing the movement costs of terrain.
Roads can be built after discovering The Wheel. With roads, all spaces cost 1/2 movement points (MP) to enter from other road tiles. After discovering Engineering, the cost goes down to 1/3 MP. In practice this means that a unit with just one movement point can move three tiles using roads. Similarly, units with two movement points can move six tiles.
To build railroads, the civilization first needs to discover the Railroad technology and have an access to Coal and Iron. Railroads are similar to roads but they reduce the number of movement points needed to enter a tile from another railroad tile to 1/10 MP.