Usually, when people talk about "graduate school" in Biology, they are talking about PhD programs, and sometimes about Masters degree programs. These are degrees that prepare you to be an academic scientist, performing research and/or teaching at a university, college, research institute, or in the biomedical research industry.
More information on Health Professional degrees, including medical doctors, dentists, nursing programs, etc.
Graduate school is not just a continuation of college as you know it - you will take a few classes, but most of your time is spent on your research project. Self-motivation is essential.
Thankfully, graduate school is not another major financial investment - most grad students in the sciences are supported with both a full tuition waiver and a stipend (currently averaging around $25,000 a year) to cover living costs (housing, food, etc.). This financial aid is either provided by the institution (through a research or teaching assistantship) or by a national fellowship. In fact, most institutions do not take on students they cannot support financially. Essentially, this means that, for qualified students, graduate school, unlike medical school, is free.
Research is the focus of all graduate programs in the sciences, and programs do not want to take on students who don't know what they are getting into. Having research experience is critical both for getting in, and for being successful once accepted.
More information on research opportunities
Grades: Criteria are not usually as strict as medical schools in terms of which courses you need to take in college, but you should take a rigorous set of science classes, and should have at least some courses in the area of study you are thinking of pursuing in graduate school. Most successful applicants to PhD programs in the Biological sciences have GPAs of at least 3.3 - 3.5, or higher, depending on the quality of the program (but these numbers are only a rough guideline).
You will need to take the general GRE, a standardized test similar to the SATs with a math and verbal section, and usually also a subject test in either Biology or in Biochemistry, Cell, and Molecular Biology. Your scores are good for 5 years; just be sure to take the test in time to have your scores ready when applications are due. There are many ways to prepare: classes, prep books, practice tests, and websites. The Career Development Office is a good place to start in thinking about preparing for these exams.
You will need recommendations. Get to know faculty early - research advisors, academic advisors, and any professor who has taught you in an upper level science course are good choices to write your letters. Make sure to keep in touch with all possible recommenders and keep your options open.
Once prepared, you will need to start finding programs that fit your interests.
More information about finding specific programs in your field of interest.
Much as with undergraduate applications, make sure to apply to some "safety" and "reach" schools along with those you think are on target for your qualifications. Four should be about bare minimum, but some people may choose to apply to nine or ten.
Applications can typically be filled out online - just be sure to pay attention to due dates, since these can vary significantly between schools.
Once they have reviewed your application, the school may invite you for an interview. Make sure to tour not only the campus but the surrounding area - you will be living there for 2-6 years or more!
The advisor is crucial to your program decision - make sure there are a number of advisors at the institution with whom you would potentially like to work.
Talk to the other graduate students and get a sense of their general mood and what their daily life entails.
Some programs emphasize coursework more than others - see how each school compares and decide if you prefer programs with more coursework or less.
Assess the overall quality of the program and if its specific strengths best match with your specific interests.
The Leadership Alliance publishes a nice booklet with more details about preparing for and applying to graduate schools in the sciences.
The Career Development Office has a number of helpful resources about applying to graduate school, although these may not be specific to each discipline.
Talk to your Faculty Advisor!