In Texas, possession requires that a person have care, custody, or control of the contraband. If you have it on your person (pocket, in the hand, etc.) that is clearly possession. Cases, especially drug and weapons cases, get more complicated when the item is found in a house or car to which several people have or have had access, or even better yet, where several people live. (For it to be a criminal offense, the person must knowingly possess the contraband.)
In cases like that, the State has to prove an "affirmative link" between the person charged and the contraband. The classic example is a bag of drugs found somewhere in a car with 4 passengers. In reality, if no one comes clean and admits that the dope is theirs, everyone in the car may get arrested and charged. But before they can be convicted, the state must show enough affirmative links to show that the person knew that they had care, custody, or control of the drugs.
Common examples of affirmative links can include proximity (who's closest to the dope), ownership of the vehicle/home, location (glove box or console in a car or someone's bedroom closet in a home), fingerprints on the baggie, officer sees one passenger reaching toward the location where the dope is found, etc.
This is the Texas definition, and it might vary from state to state. Also, based on your question, I am assuming that your question relates either to drugs or weapons, since those are the major possession crimes, but if it is something else, the rules may change somewhat.
As always, this is not legal advice; it is an answer to a question asked on a public website. This does not make me your lawyer, and nothing discussed here is privileged. You should always consult face-to-face with a lawyer before you act on anything you see on a website.
I agree with the previous answer. Possession is defined as the care, custody, control, or management of an item.
Each state is entitled to define their law as they see fit, so possession is likely defined in different terms in each state.