For some kinds of questions there are special presumptions or forms of evidence that are specified by statute. For example, in Canada and the United States, registering a copyright creates presumptive proof that the copyright exists and is owned by the registrant. In Canada, breathalyzer results are conclusive proof of the blood alcohol concentration if certain conditions are met.
Some of these regimes are not subject to challenge: they are legal facts even if they are not objectively true (e.g. the breathalyser results). Others create presumptions that can be overcome by contrary evidence (e.g. copyright ownership).
The default: present relevant evidence, including testimonial evidence
But outside of special regimes, you prove a fact by introducing evidence. The default is that all relevant evidence that bears on a material fact is admissible. Evidence is relevant when, if is were to be accepted, it would make the fact in issue more or less likely to be true. Evidence comes in many forms: testimony of the parties or witnesses, documentary evidence, physical evidence, expert opinion evidence.
If you are wondering, "How do I prove X?" Ask yourself, why do you believe X? Or, how do you know X? Or, how has the event X left its mark on the world? Whatever has led you to believe X probably is the evidence that you would want to introduce to the court to help prove X. Perhaps you saw X: you can tell the court you saw X. Perhaps you took a photo of X: you can present that photo in court. Perhaps you have a receipt for X: you can show that receipt in court. Perhaps X is a proposition about your own actions: you can tell the court about those.
Obtaining the evidence
Evidence can be obtained from the other party during the discovery process, or from third-parties using subpoenas, subject to objections relating to relevance or privilege.
Not all (potentially) relevant evidence is admissible
However, some evidence will be inadmissible despite it being potentially relevant. I will only present a few categories, some very general and some more specific:
- hearsay (unless it falls within an exception to the hearsay exclusion),
- privileged material (unless it is a case-by-case privilege or a discretionary privilege and the person seeking to admit the evidence demonstrates to the judge that it should be admitted),
- sexual history evidence of a sexual assault complainant when it will be used to support an inference that the complainant is more likely to have consented to the sexual activity that forms the subject-matter of the charge or is less worthy of belief (this is because it has been recognized as not relevant for this purpose),
- material that was obtained in contravention of the Charter and where the court has decided that the remedy for the Charter violation is exclusion of the evidence (in the United States, there is stricter, exclusionary rule),
- there are many more.
There are also some meta-rules about the evidence that may be used to impugn a witness's credibility, themselves at trial to provide evidence, but for now I am leaving those out of this fairly summary answer.
Weighing the evidence
The trier of fact (the judge or a jury) then is to weigh all the admissible evidence, including by weighing the witness and party testimony according to its credibility and reliability after testing through cross-examination, to come to a conclusion on the ultimate question(s) at issue.