A waterlogged garden is generally described as one where the soil is compacted to such an extent, that water does not run through it freely.
Because of this rainwater will remain in the soil for long periods.
This prolonged saturation leaves no air spaces in the soil which is not conducive to good growing / planting conditions.
Few plants can survive under these conditions unless they have special roots that are adapted to acquire oxygen.
Plants such as flag irises, marginal aquatic plants and willows are examples of these, whereas most other plants will literally drown and die off after a short period in these conditions.
Checking if soil is waterlogged;
The normal test to establish if a soil is waterlogged is to dig a straight sided pit about 600mm (24") deep.
This allows you to check root depths and the make up of the soil strata.
Most plant roots grow within 500mm (18") of the surface, so if adjacent roots are much less than this, then there is a potential problem.
Similarly this will determine if the subsoil is extremely hard.
To test permeability leave the pit covered overnight to exclude rain, and if water collects in the excavation, you have a high water table.
If there is no sign of water, flood the pit and leave covered as before for 24 hours, if water remains, then the drainage is poor.
If one or all of the above situations are prevalent then remedial action will be required.
A couple of solutions are;
1) Double dig the plot, and dig in horticultural grit and organic matter to loosen the soil and improve the drainage.
2) Strategically placed land drains to take surplus water away from the area will also help.
As an alternative you can form raised beds.