It is my understanding that it is incredibly difficult to be killed in Beis Din - so much so that one that kills once every seven (or seventy) years is a murderous one (Makkos 1:10). What exactly are the requirements that must be fulfilled for a person to be killed?
Chabad.org summarizes the requirements Judaism and Capital Punishment
However, Jewish law mandates many caveats to actually securing an execution. First of all there have to be two witnesses to the actual crime. Even strong circumstantial evidence is inadmissible for the purposes of capital punishment. Secondly, the witnesses need to give an official ‘warning’ to the perpetrator that if he commits the offence he is liable to the death penalty. The perpetrator needs to clearly accept this warning and commit the crime within a few seconds of the warning, thus distinguishing the act as utterly premeditated and the sinner completely aware of the consequences. Even if all the above conditions are satisfied there is still no certainty of conviction as the witnesses are subject to very aggressive cross-examination. If certain details of the witnesses testimonies do not correspond or if they are inaccurate in describing the full cirmcustances surrounding the event they can be disqualified and the conviction will not be secured.
Furthermore, according to the Talmud, “any doubts in a capital crimes should always be for the benefit of the accused” (Baba Batra 50B, Sanhedrin 79A). A judge who had argued initially for condemnation can subsequently argue for acquittal, but one who had argued for acquittal cannot not later argue for condemnation. Acquittal in capital cases required only a simple majority of one vote, condemnation required a majority of two.
A ‘guilty’ verdict can be reversed to acquittal if errors are revealed, but new evidence to reverse an ‘innocent’ decision to ‘gulity’ is not accepted!
Note that a unanimous declaration of guilty forces the court to let the perpetrator go free because any such case means that not all the facts have been revealed. A judge is not allowed to vote innocent in order to prevent such a unanimous verdict, if he believes the perpetrator guilty.
A court does have certain extra-judicial powers to kill a murderer under certain circumstances and the king can declare the perpetrator subject to the death penalty as a mored bemalchus (guilty of treason).
A detailed translation of Aruch Hashulchan Choshen Mishpat gives the details but is too long to summarize here.
As an example we have The Blood of Thy Brother
The Ohr Samayach in Hilchos Melachim Perek 3 explains that since there is some evidence that the relative was murdered this is sufficient to exonerate the Goel Hadam. The Ohr Somayach compares this to the judicial powers of a Jewish monarch. The Jewish king is allowed to kill someone who rebels against him without the requirements of Bais Din such as two witnesses and a warning. The Jewish king can dispense with these requirements to kill one of his subjects when required. The Ohr Somayach explains that this is because the Jewish king is not punishing the perpetrator but rather he is establishing the authority of a monarch. Respect for the king’s authority uses other extra judicial standards. Similarly, as explained above because of the importance in maintaining the proper respect for the sanctity of Jewish life the extra judicial methods of Goel Hadam or Kipah are used by Bais Din.
The Ketzos in Siman 2 on a theoretical basis points out that although we currently do not have a Bais Din with the authority to administer capital punishment, however, we could perhaps allow a Goel Hadam to avenge the death of a relative. Since in the case of a Goel Hadam it is not a regular punishment but an extra judicial procedure, this could be applicable even when Bais Din itself has no authority to administer capital punishment.
@MonicaCellio has pointed out that the wikipedia article Stringencies of evidence in capital cases actually has the correct list.
It requires two witnesses who observed the crime. The accused would have been given a chance and if repeated the same crime or any other it would lead to a death sentence. If witnesses had been caught lying about the crime they would be executed.
Two witnesses were required. Acceptability was limited to:
Adult Jewish men who were known to keep the commandments, knew the written and oral law, and had legitimate professions;
The witnesses had to see each other at the time of the sin;
The witnesses had to be able to speak clearly, without any speech impediment or hearing deficit (to ensure that the warning and the response were done);
The witnesses could not be related to each other or to the accused.
The witnesses had to see each other, and both of them had to give a warning (hatra'ah) to the person that the sin they were about to commit was a capital offense;
This warning had to be delivered within seconds of the performance of the sin (in the time it took to say, "Peace unto you, my Rabbi and my Master");
In the same amount of time, the person about to sin had to:
Respond that s/he was familiar with the punishment, but they were going to sin anyway; AND
Begin to commit the sin/crime;
The Beth Din had to examine each witness separately; and if even one point of their evidence was contradictory - even if a very minor point, such as eye color - the evidence was considered contradictory and the evidence was not heeded;
The Beth Din had to consist of minimally 23 judges;
The majority could not be a simple majority - the split verdict that would allow conviction had to be at least 13 to 11 in favor of conviction;
If the Beth Din arrived at a unanimous verdict of guilty, the person was let go - the idea being that if no judge could find anything exculpatory about the accused, there was something wrong with the court.
The witnesses were appointed by the court to be the executioners.
As a result, convictions for capital offense were rare in Judaism.