Mowing your lawn in the rain is not recommended. The moisture makes the grass stick together, possibly leaving clumps in the yard and clogging your mower. In addition, the mower tends to push over wet grass rather than cleanly cut it, making the mowing somewhat ineffective. That being said, sometimes it makes more sense to mow grass in the rain than to let it grow out of control, especially if the forecast shows no chance of sunlight anytime soon. If you have to mow the lawn in the rain, treat it more like a trimming than a mowing.
Prepare the lawnmower. Sharpen the blades to make cuts as cleanly as possible. Remove any grass catchers or mulching equipment so the mower will dispense the cut, wet grass back onto the lawn. Adjust the height of the blades to the highest setting.
Mow using your normal travelling technique -- do not push the mower back and forth to try and cut more of the wet grass. Walk slowly and adjust the speed setting to spin the mower's blades slowly. Turn off the mower if it clogs and allow it to come to a complete stop before attempting to dislodge the wet grass.
Aim the mower's discharge to keep the majority of the cut grass on the already mowed portions of the lawn. Mow the yard again later in the day to break up the clumps of cut grass left by the first pass. Mow frequently over the next several days if the rain is supposed to continue.
Rake the yard after the rain stops. Collect the clumped clippings for compost or disposal, or scatter the clumps into the lawn to use it as a natural fertiliser. Alternatively, let the clumped grass sit and dry for one or two days, then mow the lawn again to disperse them.