Chances are you've got heavy soil (as in clay), bad drainage, poor air circulation, compacted soil surface, the wrong kind of grass or some combination of these attributing factors.

I doubt there is much you can do about your soil type without great expence and aggravation, but you can help improve the other situations.
Once the area has dried a little give the entire site a short, hard cut. Follow this action by raking (or scarifying) in opposing directions. This opens up the surface area and lets more air in and around the remaining grass plants whilst removing some of the problem moss. Things will look messy before they get better.
Next, apply some quality grass seeds. Use a fine leaved fescue and bentgrass mixture. The smaller the leaf the less sunlight required by the plants to do well (ryegrass rarely looks good in the shade for this very reason). This will help fill in the areas left by the moss. Rake the area again if possible and then lightly roll or tread down to get the new seeds into contact with the soil.
Do not apply fertilizer as this will only encourage the exisisting grass to outgrow the new seedlings and keep the treated area cut reasonably short for the same reason (1-2 inches).
During the summer you can drive a garden fork deep into the surface at regular intervals. Fill the holes created with fine riversand or just leave them open. This will certainly help with winter drainage and surface aeration.
Not exactly a one time fix I'm afraid as the area will require a repeat treatmant every season to keep your grass free from moss.
Good luck