NewAtThis,
A short (and maybe not completely accurate) explanation of cycling: Fish produce ammonia. Ammonia kills fish. Certain bacteria will change ammonia into nitrites, which are also toxic, and then change the nitrites into nitrates, which aren't toxic, unless their concentration is too high (one of the reasons for regular water changes in an established tank is to remove some of the nitrates). So, there's the 'cycle'; ammonia -> nitrites -> nitrates.
'Cycling a tank' means establishing a bacteria colony that will be able to cycle the ammonia your fish produce into nitrates. There are a few ways of doing this, the two basic ones are called "cycling with fish" and "fishless cycling". There are pros and cons to all methods, and I think it comes down to researching them and making your own decision. I'd recommend searching for "cycling", "fishless cycling", and "new tank syndrome".
Sorry to say this, but it sounds like it may be bad news for your platys. With a new tank, I'm sure everyone will recommend getting testing kits (especially for Ammonia, Nitrites, and Nitrates), and keeping an eye on these parameters for a while.
Oh, and I can say from experience
dead fish don't always float.