When the Emperor Created the Primarchs, he Implanted them all with certain of his own Charateristics. So a "part" of the Emperor was given to each of the Primarchs, with the exception of Horus, who was made to be the Emperors succsessor.
The First space marines was made on Terra, tens upon tens of thousand. As far as I know, they had no real gene seed. But as they found the Lost Children of the Emperor, they each got 10.000 marines, and they were implanted with Gene-seed of their new primarch. So inheriting their Primarchs traits. Such as Angron's Fury, Fulgrims Craving for Perfection, The Stern Will of the Death Guard.
But the Gene-seed is more. A gene-seed also contains the experiences of previous marines. So a good first captain have a strong Gene-seed and one would be lucky to inherit it. While most gene-seeds are of Line warriors still. At least that is how I picked it up from the science talk in the HH series.