I'm in a graduate biophysics program and biophysics is a very, very broad field that different people will define differently. In general, there are two broad (non exclusive) definitions of biophysics:
1) Using and developing tools from physics to study biological systems. Things in this category include medical imaging, structural biology, and single molecule biophysics.
2) Approaching problems in biology in a quantitative, theoretical manner. Fields under this category include systems biology, bioinformatics, and computational neurobiology.
Biophysics can span very different scales. Some people study biomechanics and look at the motions of organisms. Some people study the mechanics of cells and tissues. Some people build complicated microscopes and other instruments to examine nanometer and smaller features of cells and biological molecules. Biophysics can also vary in the amount of biology involved. Some people do computational work and never set foot into a wet lab. Others are in wet lab all the time and could easily be mistaken for cell biologists or biochemists.
People in my biophysics program are mainly physics majors with varying degrees of knowledge of biology (some come in with no knowledge of biology, and some are physics/biology double majors). Biophysics programs also have significant numbers of chemists (mostly physical chemists and biochemists), engineers, and biologists (usually with some training in a quantitative field such as physics, chemistry, math or engineering). I personally majored in biochemistry, minored in math, and took many physical chemistry courses. My roommate (also a biophysics grad student) majored in physics and had almost no knowledge of biology (just a few introductory courses) prior to starting grad school.
Biology is becoming an increasingly interdisciplinary field. While there is still a need for people with traditional biology training to become experts in particular biological systems, people coming to biology from other fields with new tools and approaches are revolutionizing biology. For example, x-ray crystallographic of proteins, high-throughput DNA sequencing, and nuclear magnetic resonance imaging are a few of the technologies that have really helped to expand the questions that biologists can ask and answer.
Biophysics is certainly a rich field for physicists interested in studying biological systems, but it is not for everyone. Biological systems are enormously complex and messy, so they often lack the simplicity and elegance of systems studied by most physicists. It is very, very difficult to conclusively prove anything in biology. Furthermore, whereas physicists are used to being able to do some math to figure out how a system will behave, you would have to rely mostly on experimentation in biology (and experiments will often give confusing often contradictory results because, in many cases, your experiments will literally have a mind of their own). That said, biophysicists hope to advance our understanding of biology to where we can have good, quantitative models of biological systems. We now have great tools to study biological systems quantitatively, but biology sorely need more physicists with the training to know what to do with this quantitative data.