Planting on a Prepared Seed Bed (Bare Soil)
Planting on a prepared seed bed is the best choice. This is because you have removed a lot of competition by killing any existing "adult" weeds or other plants. Little seedlings grow better when they don't have to compete for food, water, and sunshine that an existing plant would also want and basically take first (being the big bully). Secondly, a prepared seed bed has reduced compaction of the soil allowing the roots to push through and grow (find water and nutrients) easier and faster. Third, a prepared seed bed usually ensures that "seed/soil" contact is better. The only drawbacks are having to till up an existing lawn area (perhaps killing an existing lawn) and the fact that you have bare dirt for a period of time until the new grass can establish.
Planting in an Existing Grass Lawn
Planting seeds into an existing lawn is somewhat harder to obtain good germination because seed/soil contact is often less. The key is to somehow increase the seed to soil contact. Cool season grasses usually work better (germinate easier) with this method that will warm season grasses. Aeration (using some type of spike or plug aerator, or a vertical mower to produce slits in soil) prior to seeding helps increase the likelihood of individual seeds falling next to soil and hopefully being slightly covered by soil due to wind/rain action. Most often experts recommend higher seeding rates because a certain percentage of the seedlings are going to fail, either due to competition from existing grass or the inability to properly germinate due to seed/soil contact problems. Even raking the area before or after broadcasting seeds can help increase germination and create better seed/soil contact and/or coverage.
Broadcasting on Top of the Soil/Lawn Area
This method is dependent on the amount of seed/soil contact that can occur and also on the type of grass planted (cool season/warm season). Warm season grasses will RARELY germinate if they simply are laying on top of the ground, unless some action (walking, rain, raking, etc.) causes them to become covered by soil. Cool season grasses on the other hand may partially germinate as they will occasionally germinate with less coverage/soil contact. In fact, ryegrasses have the ability to easily germinate with just good contact with the soil. Fescues and Bluegrasses require more contact for good germination to occur. Again, any method used--aeration, raking, covering with top soil, etc.--helps increase germination. With warm season grasses (Bahia, Bermuda, Zoysia, Buffalo, or Centipede), don't expect germination to be very high without good seed/soil coverage at correct depths.
Planting Early in Season or Out of Season
If you decide to plant too early or too late, perhaps weather giving you spring fever prematurely, or waiting too late in the fall to plant, don't expect germination to be as good (or perhaps NOT at all). First seeds require specific planting temperatures to germinate. Secondly, they require a certain amount of growing time at proper temperatures and conditions (rainfall, soil nutrients available, etc.) to grow to a mature enough size to survive any dormancy caused by either high or low temperatures or other weather extremes. Grass seedlings are like babies--you don't put babies outside in temperatures below 55 degrees or above 80 degrees without some additional protection to survive. There are "dormant" seeds in some grasses that are natures way of preserving the species. These seeds lay successfully dormant while waiting for the correct temperatures and weather conditions to occur before beginning germination. Often these seeds are one that is called "un-hulled" seeds. Removing hulls on seeds results in quicker germination, thus breaking natures dormancy protection.
Same Goes for Your "Baby" Grass Seedlings
Plant at the right time of year for your type of grass (early spring/early fall for cool season grasses, late spring/early summer for warm season grasses), and plant when temperatures and weather is appropriate. You can in some instances plant outside normal times, but just be aware that the risks are higher. Either rot of seeds from excess cold/moist soil can occur, or freeze/frost kill by late cold temperatures after seeds germinate due to an early warm up.
Planting Over Snow
Cool season grasses (Perennial Ryegrasses, Fescues, and Bluegrasses) can be successfully planted on top of snow for winter planting. What happens is that the seeds are carried down to soil level by melting snow in the spring. At that times cracks are present in the soil from the freezing temps of winter that the seeds can "melt" into. This provides for reasonable seed/soil contact. The disadvantages are the same as for planting out of season above; a higher risk that this method may have some failures.
Planting with Erosion Mats
Seeds can be covered by "erosion mats" that help prevent erosion. However for best results, the seeds still need to have good seed/soil contact for germination to occur. Simply covering the seeds with an erosion matting material does NOT ensure better germination. The mats are to help prevent erosion (and thus washing away of seeds), not to increase actual germination.
Modern hydroseeding methods, when correctly done help maintain good soil moisture in the top soil. The hydroseeding machine also slightly forces the seeds (by pressure) into the top soil to be covered by soil, thus ensuring better germination. Some seeds also benefit slightly by the soaking inside a tank of water prior to planting. Usually you also get a more uniform application of seeds when mixed correctly and applied correctly. This method should be only done by a qualified applicator.
Planting with a Turfgrass or Slit Seeder
Commercially designed "turf" seeders are available for planting grasses. Some are built to plant in prepared soils (Brillion seeders). Others have what are called slit discs (slit seeders/oo-till & grain drills). The discs slice very narrow holes in your soil/lawn and direct the seeds through tubes to drop into these slits, thus ensuring better seed/soil contact. They are excellent devices for seeding into existing lawns in order to improve either the variety planted or the total stand. These planters are available from a small walk behind models to large tractor types.