It does depend on the drug you are referring to, which agency arrested them, and what state the person was charged in. Possession means simply that: you possessed some illegal drug on your person or property. It could be as small as a few marijuana leaves and still be possession. They likely wouldn't charge you for such a small amount, but they legally can. The larger the amount, the more likely a person would be charged with trafficking, or possession with intent to deliver. It's the difference, legally, between drug users and drugs dealers, and the state treats drug dealers much more harshly.
How much is defined as trafficking depends on the drug in question. With cocaine I believe it's anything over 19 grams, for example, whereas with marijuana it would be a larger amount.
Laws in regards to illegal drugs vary by state. In addition, illegal drugs fall into different categories. For example, the possession of marijuana usually does not carry as heavy a charge as possession of cocaine. It may also depend on what the intent of the possession of the drugs was. If a person had drugs in their possession with the intent to distribute them or sell them it may be looked upon differently than if it was for personal use.
Punishment for the possession of illegal drugs can range anywhere from a fine to imprisonment. It depends on numerous factors including jurisdiction, kind of drug, intention, and amount of the drug.
I will assume that you are asking about illegal drugs. It depends on the State you are in. Some states can convict on possession charges with any amount at all. Other states say that the amount possessed must be a "usable amount". Usable amounts are probably left up to interpretation. Generally a "trace" amount of an illegal substance coincides with an amount that is not "usable". Some sort of residue is a good example of this. The vast majority of the States tend to adhere to the "any" amount rule, a conviction can be sustained on absolutely any amount. Two states that say the amount possessed must be a usable amount are California and Florida.
A study completed in 1997 by the Argonne National Laboratory showed that 78% of the $1 dollar bills in circulation in Chicago, Miami, and Houston were tainted with microscopic traces of cocaine.
This depends, of course, on what drug you are talking about and in what state. But generally, the possession of a small amount of a drug is a crime, but only a misdemeanor and not a felony. This is true as long as the drug appears to be for personal use.
For example, in California, the possession of one ounce or less of marijuana is a misdemeanor, as long as the drug is intended for private use. Possession of larger amounts is much more likely to get a person charged with a felony for being a drug dealer.
You can find a rundown of state by state laws at the link below.