As part of my duties in my department I have undergraduate advisees. Lots of them. Around this time of year many of the senior students are completing their graduation audits and have to meet with me. A popular topic during these meetings is "How do I get into grad school".
Now if you are just asking that question at this point in your academic career and you want to apply for grad school right after your undergrad, there is not a whole lot you can do that is going to make you stick out as an applicant that isn't already in the books. That is not to say you can't get into grad school if you decide late in the game, rather that you won't have much time to enhance your CV.
So what do grad advisors look for in the applications they receive? That depends on the type of school that you are applying to and whether or not they have a rotation program, etc. If you are applying to a program, then your stats (GPA, GRE) are going to be the main determination of whether or not you get in. My experience, however, mostly lays with the lab recruiting model, where PIs make the admission recommendations based on the individuals they would like to join their lab. In this case, PIs are evaluating the pool of candidates with the specific intention of accepting someone into their lab. Both the rotation and direct systems have pluses and minuses which are not the point here.
As someone who directly recruits, I am probably looking for slightly different things from someone directly out of undergrad. I want to know what they did that got them interested in research.
Unless a candidate's GRE scores are notably bad, I pretty much disregard them. GPA is also fairly useless, as a lower GPA in important classes can be masked by A's by a minor in Underwater Basket Weaving. I look at the transcript and check how a student did in courses related to what they need to know for the work we do. But all of that is pretty basic.
More importantly, in an ideal world I am looking for evidence that a candidate has gotten involved in actual science. Sometimes this is as simple as seeing research credits on a transcript, but even this can be deceiving because "research" in some labs is equivalent to washing dishes. Also, a lot of students don't take research for credit, getting paid or
exploited simply experience, instead. Even a semester at a field station where science is taken outside of the classroom can be some indication that an applicant has gotten their hands dirty, so to speak.
So the reference letters, to me, are extremely critical when I am evaluating an applicant. I want to know whether a student has been involved in a lab and whether they showed any aptitude for the work. There are plenty of students who are great in the classroom but just can't cut it in a lab environment. They might be great for some branches of scientific research, just not the one I live on. I want to know what the referees think of the student's potential in the lab based on their past observations. A reference from a classroom prof caries much less weight for me than someone who has seen the student work and interact in a research environment.
Once I have screened the applicant pool for students that have gotten involved in research, have demonstrated an aptitude for it and have the academic background they need to succeed in my lab, those go on the top of the pile and I work back from there until I get a group I am comfortable interviewing. I often end up interviewing candidates who don't have research experience, so it is not as though it is an inflexible selection criterion.
One of my biggest fears in bringing in an individual that will upset the fairly happy lab culture, so interviews are another critical step for me. I have seen a lab group really torn apart by a single individual, and my main focus of an interview is to do everything I can to ensure an applicant would add to, or at least not be a negative influence on, my group.
By the time everything is taken into consideration it is generally pretty easy to narrow things down to the people who I want to focus my recruiting effort on, and so far this "system", for lack of a better term, has worked out well.
Oh, and DO NOT start your personal statement with a tale of your childhood. I'm not kidding. Don't.