Although Andrew and his brother Peter were two of the original apostles, very little is known about Andrew. He was a fisherman from Galilee, who went on to spread the Christian religion in Greece and Asia Minor. He is believed to have been killed by being crucified by the Romans on a diagonal cross in Patmos, in Achaia, Southern Greece
Some 300 years after his death, the Emperor Constantine was going to move the saints bones, and legend has it that a monk (called either St Rule or St Regulus) was warned of this in a dream by an angel, who told him to remove the saints bones to the "ends of the Earth" to keep them safe.
St Rule managed to get a tooth, an arm bone, a kneecap and some fingers before setting out on an epic journey, ending in shipwreck off what is now known as St Andrews.
A chapel was built to house them, and by 1160 a cathedral. St Andrews was the religious capital of Scotland, and the goal of many pilgrims
When the 1320 Declaration of Arbroath was sent to the Pope, St Andrew was named as the patron saint of Scotland. He is also Patron Saint of Greece and Russia.
The saints remains have now disappeared, probably destroyed during the Scottish Reformation, when the strictures of Calvinism tended to wish to remove traces of Catholic "idolatry". The site of the relics is now marked by a plaque in the ruins of the Cathedral in St Andrews.
Interestingly, some of Saint Andrew's bones were taken to Amalfi in Italy. From there the church sent some fragments in 1879 to Scotland. And in 1969, Pope Paul VI gave some further relics to to the Catholic church in Scotland during a visit to Scotland
Saint Andrew's Day is celebrated on 30th November each year.
See here, for a detailed history of the town of St Andrews.