I'm not a soil scientist, but no bona fide ones have responded yet so I'm going to chime in based on what I've learned, read and observed. This is the Internet after all, if it weren't for unqualified opinions we wouldn't have much at all.
Soil is a mixture of four substances: inorganic material, organic material, water, and air. Inorganic material is ground up pieces of rock, depending on the size of the pieces it's classified as sand, silt or clay, with sand having the biggest pieces and clay the smallest. Organic material is ground up pieces of things that used to be alive, mostly plants. Those ground up pieces have spaces between them, that space is filled with either air or water. Water is important because it gives soil cohesion, it helps it stick together. Soil with no water is called dust; soil with all of the air replaced by water is called mud.
Soil that allows water to pass through it readily is considered well-drained. Well-drained soil has larger spaces between the particles. The spacing of soil is determined in part by its makeup. Generally the larger the inorganic particles the better the drainage, and the more organic material the better the drainage. It's also determined by its structure, how densely packed it is.
Poorly drained soil can be improved by adding larger organic particles -- sand-- and by adding inorganic content. It can also be improved by improving the structure. The best way to improve the structure is to get stuff to grow in the soil. Plants will send out roots that will create openings in the soil, as will having worms or insects living in it. Plants can be very useful for drying soil, the roots absorb water out of the soil that is released into the air by the leaves through a process called transpiration. Grass is particularly good at this, a healthy lawn can remove through transpiration water equivalent to a quarter inch of rain a day. Soil can also be mechanically aerated.
The big thing to avoid is soil compaction. There is a death spiral that happens with mud and soil compaction. When soil is wet the water acts as a lubricant and makes the soil easier to compact. When the soil is compacted the spacing in it is compressed and it absorbs and passes water less readily. When it rains, water is more likely to puddle, and the compacted spots often provide depressions that hold water. So the soil stays wetter longer and becomes more vulnerable to compaction. At the same time compaction kills the roots of plants, and most plants can't live in muddy soil, so the plants die. Less water is removed through transpiration, so the soil is wetter longer, and the organic content of the soil drops, further reducing drainage. The worse the drainage the worse the mud, and the worse the mud the worse the drainage. So you really want to avoid compaction, stay off your soil when it is wet and don't work it wet.
Finally, you might read that compounds containing calcium -- lime or gypsum -- will "break up" clay. In the western US the clay is what is called "sodic," which means that the clay particles are compounds of sodium. Applying calcium will tend to replace some of the sodium with calcium, which results in larger particles and better drainage. But for the clays found in most of the US applying calcium does nothing. (Although if your soil is acidic lime will really help the grass grow, which will improve your drainage).
