In British English, most of the normal uses of the word "transportation" in American English are covered by the word "transport" (trains, buses and planes are "public transport", etc.)
To speakers of British (and, most notably, Australian) English, "transportation" was a punishment for those convicted of crimes during the 17th, 18th and 19th centuries, generally for offences just below the level that would get you hanged. The convict was sentenced to spend a period ranging from a few years to life in a prison camp in one of the colonies, at first in America, but following the American Revolution mainly to the newly discovered land of Australia. Those involved were usually guilty of petty crimes - small-scale theft was probably the most common - but also offences which might now be considered as political; the Tolpuddle Martyrs, found guilty of sedition for attempting to form a trade union in 1834, were sentenced to transportation for seven years. Many - probably most - of those who served out their terms remained in the colonies, forming the core population of colonists in Australia in particular.
For further reading on the Australian experience in particular, I would recommend The Fatal Shore by Robert Hughes.