When you plant onions, you have a choice between seeds, sets, or plants.
Onion seeds are true seeds. They need to be started indoors several weeks before the time when they would be planted out. Onion seed is perishable so you should only use fresh seed when planting onions this way.
The advantages of starting from seed is that there's often more variety to choose from, and you're working with first-year plants which mitigates the risk of bolting. The disadvantage is that it is much less convenient than sets: you have to start seeds indoors in pots, care for the seedlings until they're ready to plant out, and then transplant them. Seeds are also cheaper than sets, assuming you don't assign cost to your time and your potting mix, and you already have the equipment you need to start and care for seedlings.
Onion sets are the bulbs from onions that were planted the year before. They were planted very thickly and didn't mature into full bulbs. Onions are a biennial (i.e. they flower and set seed in the second year), so the sets are the second year of the onion. When you put the sets into the ground, they sprout leaves and the bulb fills out.
The major advantage of sets is that they are very easy to plant and usually give a good harvest. The disadvantages are that there is a somewhat smaller selection of onion types, and they have a higher risk of bolting since they're already second-year plants.
Onion plants are simply seeds that someone else has started for you -- like buying tomatoes at the garden center. You take them home from store and plant them into the ground.
The advantages of buying plants is that you don't have to start and care for seedlings, you get to start with already started vigorous (hopefully) plants, and they don't have the risk of bolting that sets do. The disadvantages are that they are more expensive than either sets or seeds, they're a bit more work to plant than sets, and there's usually less variety to choose from than either sets or plants.
I've personally grown onions from all three methods, and I generally prefer seeds:
- I have more control over the end product.
- I have more choices for variety.
- I just plain like having something to plant and watch grow when there's still two (or three, or four) feet of snow on the ground in mid-February.
Though there is something to be said of the ease of just buying a handful of sets at the local garden center and dropping them into the garden when you get home...