Dandelions prefer the same growing conditions as most other garden perennials. Plant them in a well-tilled loam soil in full sun, weed out competing plants, water them well when nature doesn't, and they will grow collanders full of tasty salad greens for you. But, if you have insufficient time for this, they will grow in your lawn.
Dandelions are not native to North America and can't compete with healthy grass, so you must give them a hand. Don't sharpen your mower, ever - you want to tear the grass to shreds, not cut it cleanly. And, set your mower to as low a height as you can - 1 cm if possible, like the federal National Capital Commission does here in Ottawa. This saves you time, as the grass is weakened so much it grows more slowly. Your dandelions will recover much faster than the grass due to the energy stored in their roots.
Bagging and land-filling your grass clippings also helps - the slow-release nitrogen in them is of much more use to grass than to dandelions. It's more work though.
Don't water your lawn. The shallow roots of grass will dry out faster than the deep taproots of healthy dandelions. Not only that, but you'll have more white grubs to eat the grass - the beetles prefer dry soil to lay their eggs in. In Ottawa's dry summers, it's almost as effective as weeding them!
A liquid quick-green fertiliser in spring, as used by lawn-spray companies, helps too. The grass will grow faster than its roots, so summer drought is more effective at getting rid of it.
Finally, use a heavy ride-on mower, or even better drive a truck over your grass as City of Ottawa staff regularly do. This will compact the earth so that the surface roots of grass will be suffocated, while your dandelion roots will be able to reach below the compacted layer.
other notes on pesticides