The most common way to remove a lawn is to physically remove the sod by cutting it into strips with a spade or a sod cutter, rolling the strips up, and either taking them away or turning them over and letting them compost in place. The cut sod can be laid upside down in the middle of the area being removed, to create a mound, as fill in a low spot or to create a berm if needed elsewhere in the garden. Whether moving to a new spot, or composting in place, cover with a layer of newspaper (6-10 sheets thick, and overlapping) and 4-6 inches of compost, and lastly with 3-4 inches of mulch to decompose, create good soil, and prevent grass re-growing. (Note that if you leave the sod in place, or mound it, you’re turning a quick method into a slower one.)
Removing sod doesn’t work if you have a persistent perennial lawn grass like Bermuda grass. Simply removing the sod on Bermuda guarantees that it will be back, since the roots go down for several feet, and it only takes an inch of root to grow it all back. In this case, the only way to get rid of the lawn is through solarization (or herbicides)—plus several months of waiting to be sure it’s really gone.