The Bill of Rights: Amendment 6
Amendment VI
In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the State and district wherein the crime shall have been committed, which district shall have been previously ascertained by law, and to be informed of the nature and cause of the accusation; to be confronted with the witnesses against him; to have compulsory process for obtaining witnesses in his favor, and to have the Assistance of Counsel for his defense.
"In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial,"
The speedy part of the trial refers to the belief that a trial shall be held quickly so that if you are innocent it is (hopefully) not dragged out. There are some states that have such a backlog of cases that there are current law suits that are attempting to let people off of the offenses because the cases can sit open for years.
The reason for a public trial is that in a secret trial things can be abused and hidden to the benefit of the governing body, and may have had some of its roots in the backlash from the Spanish Inquisition and the French abuse of the "letter de cachet" (according to Cornell).
Additionally "it helps to assure the criminal defendant a fair and accurate adjudication of guilt or innocence, it provides a public demonstration of fairness, it discourages perjury, the misconduct of participants, and decisions based on secret bias or partiality. The Court has also expatiated upon the therapeutic value to the community of open trials to enable the public to see justice done and the fulfillment of the urge for retribution that people feel upon the commission of some kinds of crimes."
"by an impartial jury of the State and district wherein the crime shall have been committed, which district shall have been previously ascertained by law"
The selection of an impartial jury typically is supposed to be held of a cross section of the community. This is supposed to help ensure that the guilt or innocence of a defendant is decided in a fair manner with no one set of biases or prejudices swaying things to one side or the other.
"to be informed of the nature and cause of the accusation"
Basically no one should be arrested and tried without knowing what they are being charged with.
"to be confronted with the witnesses against him"
This specific piece of the Sixth Amendment has a couple of different purposes. Firstly this allows a defendant to face their accuser. It also has the effect of rendering secret "witnesses" or non-existent "witnesses" unusable in the court because they must appear before the court and the accused. Thirdly and possibly indirectly it also makes written statements to be equivalent to hearsay in legal matters.
"to have compulsory process for obtaining witnesses in his favor"
This means that the right to use the law (police etc.) to compel witnesses to appear in court when called is extended to the defense just as the prosecution has this ability.
"to have the Assistance of Counsel for his defense"
Essentially the purpose of this is to ensure that all citizens are allowed to have an attorney working with them in their case.
This clause seems to have arisen partly in opposition to the English practice of prohibiting felony defendants from having an attorney unless a debatable point of law arose during the trial.
http://nationalparalegal.edu/conLawCrimProc_Public/RightsAtTrial/RightToCompulsoryProc.asp
http://www.heritage.org/constitution#!/amendments/6/essays/158/right-to-counsel-clause