Compost is where leaves, twigs, manures, certain kitchen scraps, etc. have decomposed into excellent soil food.
You can buy it, but all commercial composts come from industrial waste. Usually it isn't so bad:
cattle manure and
wood chips. Nearly all commercial composts will contain traces of herbicides (the cows eat the herbicide treated grasses and the herbicide is still viable in the end product).
So it's one of those tricky things. On the one hand, compost is typically magic stuff! It adds loads of excellent organic matter to the soil. You can top dress your lawn up to an inch and the grass will grow up through it. Earthworms will thrive and work it into the soil (provided that manures were not used where the animals were treated with long term worming medication). On the other hand, it is nearly impossible these days to get a compost that doesn't have some downsides to it.
Let's assume that you are going to move forward with purchasing compost. It will probably contain wood chips. If it does, your lawn will green up for the first week and then suffer from nitrogen deficiency. The deficiency is caused by the wood chips. As the wood chips break down, they suck up nitrogen. Months or years later they will release that nitrogen again. If you are savvy about this, all you have to do is know it is going to happen and sprinkle a little organic lawn fertilizer a week or two down the road. The wood chips will take up some of that too, but some of it will find it's way to the grass.
I think the best thing to do for improving lawn soil is to fertilize, mow often,
water infrequently and optimize your pH.
I visited a friend last week who this year started following my advice. Last year his lawn was miserably pathetic. This year, he has the best looking lawn on the block. For him, the big difference was mowing high and fertilizing.
