The branch of Martins I belong to is of Gaelic descent. The name was originally Mac Ghille Mhartain (translates to "descendant of a servant, devotee, or follower of St. Martin) which got shortened over the years to McMartin, Gilmartin, Kilmartin, McGilmartin, McGilliemartin, McKilmartin, Martinson or Martin. In the 1600s the Scottish Clan Martin was broken (the king confiscated their lands and outlawed the clan chief) and members of the clan were absorbed into Clan Cameron and Clan McDonald.
The Irish Martin families may be descendants of Scottish Martins who settled in Ireland or may be independently derived.
French Martins are probaly mostly independently derived but some may be descendents of the Scots who fled to France after the failure of the 1745 uprising.
German and Scandinavian Martins, Mertons and Martens are almost surely independently derived. St. Martin of Tours was a popular saint and the patron saint of soldiers