For example in Russian a man named Ivan (John) whose father's name is Nikolai would be known as Ivan Nikolaevich or "Ivan, son of Nikolai" (Nikolaevich as patronymic). In women the ending is -evna or -ovna.
In Scandinavian languages, the patronymic was formed by using the ending -son (later -sen in Danish) to indicate "son of", and -dotter (Icelandic -dóttir) for "daughter of". This name was generally used as a last name although a third name, a so-called byname based on location or personal charateristic was often added to differentiate people.
In Dutch, the endings -s and -sen, for son, was used for sons and often for daughters, too. The Dutch sometimes used -dochter or -dr for daughters.
In Romanian, the endings -escu and -eanu were used, like Petrescu - son of Petre (Peter), many of the current Romanian family names were formed like this.
In many areas patronymics predate the use of surnames. They are still used in Iceland - along with the less common matronymics - where few people have surnames. Many English, Welsh and Scandinavian surnames originate from patronymics, eg. Wilson (son of William), Powell (ap Howell), Johansson (son of Johan), Eriksen (son of Erik).
See: Icelandic names