Not Medical Advice: Anecdotal sources suggest that they're still going to check you even if you're on your period. Just wear a tampon.
They see it every day, and they're not weird about it. It will be just a quick glance, then you get dressed and on to the next test.
MEPS stands for Military Entrance Processing Station. Here you will take the ASVAB, have a physical, and (if you pass) - choose your job and swear in. Everything together should take 2 days, though there are some situations that may make things shorter or longer.
Some people will go into their branch's DEP (Delayed Entry Program), and go home after MEPS to wait until their ship out date. Others will embark to basic training immediately following MEPS. Remember, you have NOT joined the military before going to MEPS, but by the end of it you will be asked to.
Check out some helpful hints for before you go to MEPS at Military.com.