Does that mean you impeach their character, or can you impeach them as a witness? What happens if the judge agrees, does that affect their testimony or their case? Can a person lose their case if you prove they have lied enough
Yes, yes, and yes. Impeaching a witness means showing that the witness is not credible. Impeachment is a very effective tool/technique and can adversely affect the other party's case, even to the point of losing. (Notice I said "can" and not "will.") There is a California jury instruction that may be helpful for you to read. CACI 5003. (Just Google "CACI Jury Instructions" and go to #5003.) One of the paragraphs basically says that if the jury believes a witness lied under oath, the jury can either disregard anything that witness said, or only those things the jury found to be false. Basically, the same rule applies if the Judge is trying your case -- that is, if the Judge is the "trier of fact."
This response is based on limited information. It is not meant as and does not constitute legal advice and does not create an attorney-client relationship.