Latin legal maxims often compress an entire rule of procedure into two or three words, and onus probandi is one of the clearest examples. Taken apart, onus simply means a load or a burden, and probandi comes from the verb meaning to prove. Put together, the phrase is literally "the burden of proving."
In practical courtroom terms, this burden decides who has to do the work of convincing the judge that a fact is true. The general rule is that whoever affirmatively asserts a fact, rather than merely denying one, carries the onus of proving it; a person cannot expect the court to simply accept a claim without supporting evidence, and equally cannot expect the other side to disprove something it never asserted.
This is different from the evidence itself, which is just the material, documents, or testimony, used to discharge that burden, and different again from the "fact to be proved" or factum probandum, which is the specific fact in issue rather than the duty to establish it. It is also narrower than the criminal-law standard of proving both the act and the guilty mind beyond reasonable doubt, which is one particular, heightened application of a burden of proof in a specific context, not the general meaning of the term.
So onus probandi simply means the burden of proof, the responsibility resting on the party making a claim to support it with evidence.