So you're interested in graduate school? Presumably in physics? I was planning on giving my own advice, but I realized other people (and some with more relevant expertise) have already given some great advice. So here is a guide to get you up to speed about the application process.
Most important advice: get lots of opinions! There is not one correct way to apply to graduate school. But if you talk to enough people, you will start to see common threads of advice.
To start with, read Philip Guo's advice on applying to graduate school. I strongly agree that graduate school is closer to a job than school as you know it.
Next, read some advice from physics professors who have been involved with the admission process. You can find the advice at Cosmic Variance Advice 1 and Cosmic Variance Advice 2.
After reading the above articles, you should have a really good idea what it means to apply to physics graduate school. At this point, you might have more specific questions. You are probably still in college (or a recent graduate), so utilize your resources. Talk to the grad students and professors in your department (and don't forget to talk to older undergrads)! They have all gone through the process and have great insights. And when you are having a conversation, it is so much easier to get relevant advice (vs general advice on the internet). Just remember, you will get lots of contradictory advice, so in the end, you have to decide who to listen to.
At this point, if you are seriously considering graduate school, I urge you to buy Getting What You Came For: The Smart Student's Guide to Earning an M.A. or a Ph.D. It will be the best $15 you have ever spent. Or if you want an overview of the book, here is a summary.
At this point, most of your questions have probably been answered. If not, check out Physics GRE Forums and GradCafe Forums. These are run by students for students. Almost all questions have been asked, but the quality of the answer varies, so take everything with a grain of salt.
So are you ready to seriously consider physics graduate school? Congrats! It is not an easy path but it can be rewarding. Here are some more useful links.
First, if you are still an undergrad, make sure you get some research experience. Either do research at your university or look into REUs.
You will have to take the physics gre. I have more on that here.
You should also consider applying to fellowships. They won't necessarily help you get into grad school, but they will help you succeed. I have some advice for the NSF fellowship.
It can be really tough to gauge what type of schools you are qualified for, so when the time to actually apply comes, check out the Physics GRE forums where people have posted their statistics. You can check out 2011 and 2010 data (and keep going back into the past if you want).
Good luck! Let me know if you have any good advice or other websites I should link to.