The process of adoption starts on a neighbor's island. They must first receive the random popup in their game telling them what they've found. They will publish it onto their feed, and everyone has a grab at it. If you are the first player to click the link, then whatever it is that you clicked on will be yours. In the case where you are not the first player to adopt the animal or crop, then you will find an in-game popup telling you that you didn't manage to adopt the animal or crop. You also receive 10 c for trying!
Likewise, any animals that you find on your island, you won't get to keep. They go onto your feed where your neighbors can make a grab at them!
See Saving Adoption Events for info on how to send an adoption link to a specific person.
Process of Adoption Visual Guide
Click the pictures to view a larger version (if available). This example uses the Seagull, but all adoptions behave the same way.
There is a chance of a pop-up appearing on your island each time the game saves, whether manually with the save button, or automatically when the save button turns gray. This is the same with your neighbors' islands.
Each person can find to a maximum, each pop-up once a day. (Keep in mind timezone differences). This means each neighbor can find 1 of each adoptable animal and the adoptable crop in a day. The more neighbors you have, the more posts there will be.
A list of adoptable animals:
Originally there was only one crop that could be obtained as a cutting from the feed, and that was the Lilikoi. As of August 2011, there is also another cutting that has been made available - the Yucca.