Water always flows to the lowest point. You have to find the lowest point away from your lawn and direct the excess water in that direction. Begin by drawing a map of your lawn that shows all the high and low areas. Water will naturally collect in dips and shallows.
A broken underground irrigation or water pipe, excessive rain and melting snow can all cause water to collect on your lawn. You should match your drainage system to the cause of the standing water.
If water is collecting under one or more low points in your lawn, you can build soakaways under those low places. Dig a hole about 2 feet deep and 2 feet wide in the areas where water is collecting. Fill this hole with stones, broken bricks or rubble, and cover it with at least 4 inches of topsoil for your grass. When it rains, the water has to soak only 4 inches before it drains rapidly into the soakaway, where it will dissipate into the soil.
You can drain the water from low spots in your lawn through 3-inch-wide earthenware drain pipes covered by gravel and with your grass on top of that. A 100-foot-long drain pipe should drop at least 1 foot. Your drain pipe should lead to a soakaway where the accumulated water will eventually disappear into the ground. Your soakaway should be at least 16 feet from your house or any other place where people are living.
The most sophisticated and expensive way to drain your lawn is to construct a central drainage system under your lawn. You lay a central pipe at least 4 inches wide. From this, in a herringbone fashion, add subsidiary pipes that are at least 3 inches wide. The subsidiary pipes should be from 10 to 40 feet from each other.
If your lawn is growing on compacted clay, building drain pipes may not be enough. You may need to use a garden tiller to remove your lawn and replant it over the clay mixed with gypsum, compost or sand.