You might know that evidence the cops find during an illegal search of you or your belongings is probably inadmissible in criminal court. You might also know that the prosecution typically can't use something you've said to the police if officers violated your rights in obtaining the statement (for example, by coercing it out of you).
Generally speaking, the prosecution can't use evidence that comes directly from police illegality—the seized object or the statement. But oftentimes, it also can't use evidence that derives from the illegality—something the officers discovered as a result of the object or statement. The latter is commonly referred to as "fruit of the poisonous tree."
Fruit of the poisonous tree includes evidence gathered from just about any kind of police conduct that violates a defendant's constitutional rights. Take an illegal wiretap, for example. Suppose the police begin to listen in on and record the statements of suspected drug dealers without first getting a warrant. One of the dealers says that he left some cocaine in an abandoned warehouse so that his buyer could pick it up. The police go to the building and find the drugs. Not only is the illegally recorded statement (the poisonous tree) inadmissible, so too are the drugs the officers found (the fruit of that tree).
In a case that developed the concept of "fruit of the poisonous tree," Wong Sun v. United States, the prosecution introduced drugs into evidence against the defendant. Federal officers had learned about the drugs from a witness they knew about only because of a statement by the defendant during an illegal arrest. The Supreme Court ruled that everything the officers discovered as a result of the illegal arrest was fruit of the poisonous tree: not just the statement itself but also the witness information they gleaned from it and the actual drugs that the witness led them to. (371 U.S. 471 (1963).)
There are exceptions to the fruit-of-the-poisonous-tree doctrine, meaning that some evidence may be admissible even though police came by it illegally.
Courts use the terms "inevitable discovery" and "attenuated taint" to describe situations in which the government finds evidence illegally but could have found it lawfully. In those instances, the evidence may be admissible. (For another way to get illegally obtained evidence into court, see Is illegally seized evidence admissible to attack a defendant's credibility?)
Consider again the wiretapping example. Immediately after the officers go to the warehouse and snatch the drugs, the police hear from a reliable informant, who reports the cocaine and its location. The court finds that the informant's tip would have provided enough information for a lawfully issued warrant to search the warehouse. In many places, a court would probably admit the drugs into evidence because the officers could—and presumably would—have found them without the illegal wiretap.
Another example of the "attenuation doctrine" occurs where an officer doesn't have a legitimate reason to stop someone but discovers that the person stopped has an outstanding arrest warrant. If the officer arrests and searches the person, there's a good chance that any evidence the officer finds will be admissible in court. (For a detailed explanation of this kind of situation, see this article on the attenuation doctrine.)
Another important exception to the fruit-of-the-poisonous-tree rule involves statements by defendants. If officers beat a statement out of a defendant, both the statement and evidence it leads to are inadmissible. But if the defendant gives a statement voluntarily, albeit without the requisite Miranda warning, evidence the police locate because of that statement can come in at trial. It doesn't matter that the statement itself is inadmissible—the poisoned fruit is nevertheless edible. (For more on Miranda and its exceptions, see When Police Violate the Miranda Rule and Exceptions to the Miranda Rule.)
Just like most legal concepts, the fruit of the poisonous tree doctrine is complex, with nuances and exceptions. If you face criminal charges, consult an experienced criminal defense attorney. An experienced lawyer can protect your rights and zealously defend you.