Ok.
Firstly, at your level, don't try to get too muddled in this civil/private distinction. Basically, both types refer to where A sues B. As noted above, public law involves the government, and its role in regulating society -->constitutional law is a good example.
The bottom line: civil law involves actions between citizens, be it tort or contract.
The burden of proof refers to who needs to prove what. So yes, in a civil matter (which is the better word to use), the plaintiff needs to bring a case and prove things. And if that can be established, it then shifts to the defendant to disprove it.
And the standard of proof refers to the level of proof required - "balance of probabilities" means that the judge needs to think that one side was more likely than the other to have told the truth.
This should be contrasted with criminal matters, where the prosecution (the state) needs to prove its case beyond a reasonable doubt (i.e. no doubt can exist in order to secure a conviction). As you can see, the burden of proof is higher in criminal matters, and rightfully so.
To say again - don't get hung up on this private/civil law divide, it only muddies the waters. Not now at least.