What Gyroids are
Gyroids are collectible, odd little noise-making creatures/robots that you dig up the day after it has rained or snowed.
What is the point of gyroids? We still don’t know. All they do is make annoying noises in your house. They do, at least synchronize with the beat of your radio.
By the way, Lloid at the auction house is also a gyroid.
What to Do With Gyroids
Keep in mind that collecting every gyroid can take a looooong time because there are so many different kinds. This also takes a lot of space! You really don't want to collect them in your house.
It turns out, however, that Brewster (owner of the Roost coffee shop) is really into collecting gyroids. He will store your gyroids in his back room for you if you bring them to him. Brewster will only offer to do this after he has gotten to know you. Thus, you must drink seven cups of coffee (while they are hot, don't tell him you are going to wait for it to cool). You can only drink one cup a day, so it will take you at least seven days to do this. After your seventh, put your gyroids in your pocket and go talk to Brewster (from the side of his counter, don't sit in the chair). He will offer to store them. From now on, just go to his counter with all your gyroids and he will store them. Then you can view them and pick the ones you want to take out for your house. Unfortunately, you cannot actually go to his back room. It looks like you could, since he has a door, but--no. This is what the storage screen looks like:
There is a regular, mini, mega, and tall gyroid for most sets, although some have more or less than that. For example, there is a Lullaboid, Mini Lullaboid, Mega Lullaboid, and Tall Lullaboid. Some sets have wee, squat, or slim gyroids, for instance, the Dingloid set has all the normal gyroids plus a Wee Dingloid and a Squat Dingloid. The Droploid set only has one: the Tall Droploid.
Tom Nook will pay 828 bells for every gyroid you sell to him. It does not matter what kind. Last but not least, you can use them to trade with other characters. Most of the time villagers will pay more than Nook will pay; however, it takes a lot longer to sell them because you have to find someone who is willing to buy it from you. You can also auction off your gyroids: see the bells page for auction info.
Unfortunately, you cannot order gyroids from your catalog. Once you get rid of one, the only way to get it back is if it shows up buried again. You will find that some gyroids are more common than others.