Law enforcement officers investigate crimes and collect evidence. It's up to prosecutors to get the evidence admitted in court, if possible under the rules. But that's not to say that police don't need to understand how the techniques used to collect the evidence can affect whether or not that evidence will be admissible. You do.
Hearsay rules confound police, lawyers, and judges alike. "Hearsay" is a statement made outside the courtroom that might be true or false, repeated in court to prove that it was true. For example, a witness to a traffic accident tells an officer, "The motorcyclist ran the red light." If the officer tries to repeat that person's statement to a jury to prove that the motorcyclist ran the red light, it's hearsay.
The general rule on admissibility of hearsay is that because it may be unreliable evidence, it can't come in at trial unless there's a recognized exception for the particular form of hearsay, or for the particular circumstances under which the statement was made, that establishes reliability. In most jurisdictions, hearsay in the form of business records and official records can be admitted. Spontaneous statements blurted out by someone under the stress of an exciting event can usually come in.
But while the admissibility of hearsay in a civil trial will be controlled by the applicable state or federal rules of evidence, a criminal trial has an extra test of admissibility.
The Sixth Amendment to the U.S. Constitution guarantees a criminal defendant the right "to be confronted with the witnesses against him." So even if a particular hearsay statement might be admissible under some hearsay exception in the evidence code, it will still be inadmissible in a criminal trial if admission would violate the defendant's right of confrontation.
Ohio v. Roberts
In the 1980 decision in Ohio v. Roberts, the U.S. Supreme Court said that the Sixth Amendment did not prevent the use of hearsay at a criminal trial as long as the hearsay statement came within a "firmly rooted hearsay exception" or had some other "particularized guarantees of trustworthiness."
This ruling was especially important in certain kinds of cases where victims or witnesses initially tell officers the truth, and later recant or forget what they said, such as child sexual abuse cases and domestic violence cases.
According to this ruling, a variety of statutory hearsay exceptions might make the initial statement admissible at trial, or the court could find sufficient reasons to consider it reliable (such as independent corroborating evidence), and admit the hearsay statement as proof of the defendant's guilt.
Crawford v. Washington
But in 2004, the Supreme Court overruled Roberts and held instead that "testimonial hearsay" is inadmissible under the Sixth Amendment unless two conditions are both met: (1) the person who made the statement must be unavailable for trial, and (2) the defendant must have had a prior opportunity to confront and cross-examine the person about the statement. The Crawford ruling will affect the admissibility of some statements, especially in child-witness, domestic-violence, and deceased-witness cases-depending upon how the statements are obtained.
What exactly is "testimonial hearsay?" The Supreme Court declined to give a full definition of everything the term might include, but did say that at a minimum it includes prior testimony (whether at a grand jury hearing, preliminary hearing, prior trial, or civil deposition), as well as statements resulting from "police interrogation." In other words, where a police officer uses "structured questioning" to obtain a witness' statement, that statement cannot be admitted in a criminal trial to prove that what was said was true, unless the witness is unavailable to testify and was subject to cross-examination by the defendant.
In practical terms, this means that if you question a child abuse victim, or the victim of domestic violence, or an assault victim, for example, and if that victim does not take the stand to be cross-examined (at a hearing of some sort or at the trial), then even if the victim is not available by the time of trial, the statement you took cannot come in to prove guilt, under Crawford.