You most likely have plantar facitis (sp?). It will hurt more in the morning and less as the day goes on. It is caused by scarring on the tendons that run along the underside of your foot -- attaching to the ball and the heel. THis causes them to not stretch as well as they used to. As you sleep, they lose there stretch and then - presto! Serious morning arch and or heel pain and lesser pain when you stretch them or just as the day goes on. It can get worse (I get it as well). What you need is to get some really good arch supports and maybe gel insoles for your workout shoes. There are also special things you can wear at night to help maintain the stretch.
This problem did once cause me to simply have to stop running altogether for a couple of months, but the eliptical, bike, etc. can still give you a great work out -- you will not have to stop altogether. It is just possible that you will not be a runner. I have given up running, but still get plenty of good cardio. Or, since you have caught it early, you might be able to intervene now and still run after resting it fir a little while.