My new Primary Investigator, who is well respected in her field, gets really nervous when talking to me for some reason. We are the same gender and both married so it's not a problem of that sort... This does not seem to happen with any other student, who she has a really good relationship with, but when talking to me, sometimes (not all the time) she really fumbles for words. This then makes me start to get nervous and our interactions sometimes become awkward. How should I improve our relationship so I can communicate effectively with my advisor?
I would advise you to simply ask about it, in a way that pre-supposes as little as possible. For example, you might say something like:
Can I ask you about something I am feeling concerned about? I've feel like our interactions become very awkward and stumbling sometimes. When I see you talking with other people, however, I do not see this happening. I'm concerned about this, because I want to have a good advisor/advisee relationship and be able to communicate effectively with you.
Don't assume anything about her feelings---what you might be interpreting as feeling nervous might be something else entirely, even just a mannerism of your PI. You also may be incorrect that this is unusual with you: people have strong observational biases and also you don't see your PI when she's alone with other people.
Hopefully, the two of you can then have a short awkward conversation that will help resolve (or at least allow you to work effectively despite) your other awkwardness.
It is impossible to hypothesize about an exact reason with such few information but I wish to make a few observations.
Have you analyzed yourself thoroughly?
It may be that you are anxious yourself. It could make other people around you to feel uncomfortable. It may be that you are seeing the reflection of your anxiety on others.
It makes a lot of difference for me when a professor greets us with a smile before starting a lecture, rather than being gloomy with eyes of a dead fish. Try to wish her everyday with a smile.
Assuming you have already analyzed yourself, I agree with @jakebeal's answer, but my advise is to converse by email due to obvious reasons.
May be you just need to break a little ice between you and her. Invite her to a coffee sometime, may be you could talk about your family, about anything other than your subject.
A personal side note: May be its just me, but I sometime feel highly uncomfortable with piercing staring of some of my peers. I use this strategy of talking not looking into their eyes. It may be awkward but gets the job done. If you wish you could try talking by not making any eye contact to rule out if that is not the case. This is an entirely subjective observation and take it in that spirit.
I recommend bringing it up explicitly only as a last resort. PIs don't, as a rule, like to delve into interpersonal psychological issues with advisees. It has the very real possibility of marking you as high-maintenance (focusing on feelings and personal stuff rather than science) and could exacerbate future awkwardness.
Before even seriously considering bringing it up, I'd make sure of the following:
This has been going on for a long time (a couple of months, at least). That is, give the relationship time to settle into an equilibrium. It could be early relationship weirdness and it will just go away.
You have introspected honestly about the causes, and worked to eliminate any obvious causes. One thing that strikes me about the OP is the lack of any consideration of alternative hypotheses, causes, or remedies. It strikes me as a bit casual and unreflective diagnosis, frankly. This may just be the way it was written, I realize, but a lot more naunce is called for.
You have gotten confirmation from people you trust (not just friends) that she is indeed acting weird. The internet is good, but people that know both of you, and can give discreet and objective advice, would be even better. Discretion is important here: don't start trash-talking behind her back if she is your advisor. :)
It is interfering with your ability to do good science together, and with your mentor-mentee relationship in a way you find unacceptable. After all, sometimes advisers are just awkward, but they still give great professional and scientific counsel. This is actually very common, if not ideal.
If those four conditions are met, then it might not be a mistake to talk to her. The PI isn't a friend primarily, but the boss, so it's important to act accordingly. My hunch though is that if you really take the time to think through the four conditions I mentioned, you will largely resolve the problem. That's my experience as both a advisee and an adviser for many students over the years.
Could it be that the PI feels intimidated because you're really smart or knowledgeable? Even if you're "just" a student and she's a PI, she might be suffering from a bit of the "impostor syndrome".
Anyway, I suggest approaching this step-by-step.
It might be something you do unconsciously, something that wouldn't bother most people. So let's rule that out. Explain the situation to a couple of close friends (people you can trust to tell you the truth even if it might hurt your feelings) and ask them if they can think of any reason why someone might act flustered around you.
It might be something specific to the way the two of you interact, or it might even be your imagination. Perhaps you could talk to another student about this; ask if they've noticed the same thing and if they have any idea why this is happening.
Finally, it could be something specific to the PI. Maybe she is attracted to you, or you remind her of someone from her past. So it might be time to follow jakebeal's suggestion of asking her about it. However, depending on what the problem is, this could make things more uncomfortable.