DISCLAIMER: Please note that the information contained in this post does not constitute legal advice. The law is in perpetual evolution, and this information may be outdated. Furthermore, this information may have little or no relevance for your particular legal predicament. Retaining a lawyer is the surest way to obtain useful and accurate legal advice.
Beyond a reasonable doubt is when a party must prove that something happened, without leaving a reasonable doubt that something else happened, in the mind of a reasonable person. This is the standard of proof required in most criminal cases. The guilt of an accused has to be proven beyond a reasonable doubt.
Balance of probabilities refers to the burden of proof on a party that has to prove that a particular thing is more likely to have occurred than to not have occurred. This the standard of proof required in most civil cases, but it also applies to certain situations in criminal law.
For instance, mitigating circumstances (to the benefit of the accused) at sentencing have to proven on a balance of probabilities, whereas aggravating circumstances (to the detriment of the accused) have to be proven beyond a reasonable doubt. See subsections 724(3)(d) and 724(3)(e) of the Criminal Code of Canada.
Sufficiency of proof: In a preliminary inquiry the Crown must show that it has enough admissible evidence which, if accepted by the trial judge, could potentially convict an accused.
Air of reality: at a criminal trial, if there is some evidence that a defence could lead to an acquittal of the accused if it was placed before a jury acting reasonably, then the defence must be considered by the jury. In addition, the Crown has to prove beyond a reasonable doubt that each element of that defence fails.