Is it necessary to understand every detail of a research paper?

I am currently an undergraduate thinking about doing a Bachelor's thesis (and publish one or two papers if at all possible). I have found an advisor in computer science (artificial intelligence). I have begun to review literature suggested by my advisor. Most of the times I can understand the general idea of a paper (and why it is novel or important). However, very often I can't understand all details of it. Sometimes it is the lengthy math proof that I can't follow and sometimes it is the algorithm description that lacks too many details for me to code it up by myself.

So is it necessary or recommended for me to figure out everything in a paper? I can certainly ask for advisor all questions I have about a paper but I do not feel good about it. I think it is a kind of wasting his time to go through all mathematical details that support a major claim which I have already understood. Also I want whether this aspect of literature review is different if I am an undergraduate vs. PhD student vs. professional researchers.