There are a number of lawn pests that cause damage such as you describe. Usually they are grubs of one sort or another, such as those of June bugs, crane flies, etc. Bt (Bacillus thuringiensis
) is usually a good way to control them. Sometimes, though, you have to know the specific grub you're treating for, as certain strains of Bt are more effective for certain grubs.
You can use a knife ... I have a short-bladed pruning saw I use for the purpose ... to cut through the sod along 3 sides of a square. Make each side about 12". Then, just grab the grass and pull back the section of sod to examine the soil underneath. You can count any grubs you see there, and perhaps collect a sample. Make your sampling area just outside one or more of the dead areas, because the grubs eat the roots. They won't be in the actual dead areas, because they will have eaten all the roots there.
Show the grubs you collect to the specialists at your nearest Extension Service office. They will very likely be able to ID them for you, because they'll be familiar with the lawn pests in your exact location. They'll also be able to advise you on the best way to control the problem, offering suggestions for organic control or safe chemical control, depending on your preference.