Marigolds can be propagated by seed or by rooting stem cuttings

From seed

Marigold seeds germinate easily and produce strong plants. 

Fresh seeds have a germination rate of about 90–95%.  Marigold seeds can remain viable for about 1–2 years.  Marigolds are self-seeding, so their seeds will drop and spread throughout your garden. To limit self-seeding, you can deadhead before the blooms go to seed. 

From stem cuttings

Choose a healthy stem that is free of disease and doesn’t have flowers or buds 

    Cut a 4–6 inch length of stem  Remove the lower leaves  Dip the end of the stem in rooting hormone  Plant the stem in a pot filled with a mix of potting soil, sand, and perlite  Moisten the potting mix and place the pot in a plastic bag to create a mini greenhouse  Put the pot in a warm, bright area, but out of direct sunlight  Lightly moisten the potting mix every four or five days  When the stem has rooted, transplant it into a larger pot