The train between London and Edinburgh is operated by East Coast, travelling up the east side of England and Scotland along the historical East Coast Mainline, between London King's Cross and Edinburgh Waverley. The relatively quick journey is a great alternative to flying, as it gets you right to the heart of the cities in a matter of hours, and for a very reasonable price (as long as you book in advance!).
In the past, a number of these services continued to/originated from Glasgow Central. East Coast now only operates one Glasgow service per day Mondays-Fridays, none on Saturdays and only one on Sundays. If you are travelling between London and Glasgow, the West Coast Man Line is a better option.