It's a simple word, credibility. Basically, it means a person's believability. A person may not be credible because he has a history of lying, or because what he says doesn't make sense when taken with other more believable facts.
In our recent trial, which was a liability dispute with no independent witnesses (we commonly call these "he-said, she-said" cases, regardless of gender), every single lawyer used the word credibility in opening statement and closing argument. Essentially, we all argued that our clients were credible, that the other side did not testify credibly, and sometimes that the witnesses were somewhat credible. It's standard fare for a trial.
In a trial, credibility is the most important thing a witness or a party has. If the judge or jury has any reason to doubt that person's truthfulness on any single point, even something unrelated, that doubt can cast a shadow on every point of that person's testimony. That's why some lawyers will fight hard to find a lie or a mis-rembering in testimony. It can be the difference between a win and a loss.
In another recent trial, I had to inform a judge that it was okay to find that my client's recollection was not credible. There, my client was exiting a bus when the driver prematurely closed the door. My client told me at our initial meeting that the door closed on his forearm as he was exiting. The impact caused a contusion (bruise) on his arm, and it wrenched his shoulder, requiring a few weeks of physical therapy. That's all well and good for a negligence case, however, the bus video showed something different. Based on the video, it looked like the bus doors closed on a plastic bag that was wrapped around his wrist. The video was clear, however, that it instantly caused injury to his arm--he shook it in pain when the doors were reopened.
My client maintained his story--the driver closed the doors on his arm. I explained to the judge that the video shows what it shows. To the extent that there is any deviation from my client's recollection, it is understandable. He was not facing the door when it closed, so couldn't see it. The contusion on his arm could have been the result of a door hitting it, or else it could have been a result of the bag handles pulling on his arm when it was caught in the door. His instantaneous perception did not change the fact of the accident, or even the injuries. In that case, his credibility, though perhaps damaged a bit, did not affect his recovery.