Area times depth, divided by 324. The measuring is trivial. The depth is where the decision is, and going too deep does real damage to the plants you are trying to help.
Mulch goes on planted beds, so you want the bed area rather than the yard area. Beds are rarely rectangles, which is where most of the measuring effort goes.
For a bed with straight edges, measure length and width and multiply. For a curved bed, the practical approach is to approximate it as a rectangle that splits the difference, running along its average length and average width. Curved bed edges do not need precision, because the depth decision swamps any error you make on the outline.
For a long thin border, measure the length and the average width. Borders taper, so take the width at three or four points and average them rather than measuring at the widest.
Subtract anything large that is not getting mulched: a mature shrub's footprint, a rock feature, a pond. Do not bother subtracting individual plants.
The standard recommendation is 2 to 3 inches. That figure is not arbitrary.
Below 2 inches mulch does not do its job. It fails to block light, so weeds come through, and it dries out too fast to hold moisture in the soil beneath.
Above 4 inches it starts causing harm. Roots need gas exchange, and a thick blanket of mulch suffocates them. Deep mulch also stays damp against stems and trunks, which encourages fungal problems and invites rot at the collar of the plant.
So the useful range is narrow, and the difference between 2 and 3 inches is a 50 percent difference in how much you buy.
If there is existing mulch in the bed, measure what is already there before ordering. Topping up a bed that already holds 2 inches with another 3 gives you 5, which is into the harmful range. Most annual refreshes need only an inch or so, because the existing layer has broken down rather than disappeared.
Cubic yards equals length in feet times width in feet times depth in inches, divided by 324.
A bed 30 feet by 6 feet at 3 inches: 30 times 6 times 3 is 540, divided by 324 is 1.67 cubic yards.
If you prefer working from coverage, one cubic yard covers about 162 square feet at 2 inches, 108 at 3 inches, and 81 at 4 inches. The full conversion guide is here.
Bagged mulch is commonly sold in 2 cubic foot bags, so it takes about 13 and a half bags to make a cubic yard. Bulk is cheaper per unit and involves a pile on your driveway.
The crossover point is usually somewhere around one cubic yard, but it depends on delivery charges in your area and whether you have somewhere to put a heap. Below that, bags are easier to handle and you can buy exactly what you need. Above it, bags become both expensive and tedious.
Worth calculating the bag count even if you are ordering bulk, because it is a useful sanity check. If your calculation says you need 60 bags for a small front bed, something is wrong with the numbers.
Around 5 to 10 percent over covers uneven ground and spillage while spreading. Beds are rarely level, and hollows absorb more than the average depth suggests.
Do not over-order heavily on the assumption that more is better. Leftover mulch stored in a heap composts and can go sour, and the temptation to use up the extra by spreading it deeper is exactly the mistake the depth section warns about. Gravel follows the same arithmetic with different depths, and measuring the yard itself is covered here.
If you would rather just be told what to buy, our best laser measures are on the front page.