For some people, seeing a bee is just another part of nature; for others, it is a terrifying sight that evokes the immediate fear of its stinger. However, if you see a bee flying low around the grass, instead of up around the flowers, it could be for a number of reasons, some of which really should provoke caution.
While a dandelion might look like a flower if you are a disinterested observer and a weed if it is growing in your lawn, a bee simply sees a source of nectar. As a worker bee's primary responsibility far away from the hive is to forage for this food -- and pollinate flowers in the process -- if a you see a bee flying low around the grass, it could be approaching a dandelion or looking for other small flowers.
Not all bees build nests up off the ground. Some bees, such as the ground bee, build their nests underground. The ground bee, which is actually a kind of yellowjacket, converts old mole or rabbit holes into its own nest. Thus, the bee you see flying low around the grass could be either getting ready to enter its nest below ground or looking for a new nest location. Be cautious if it is a ground bee; they are aggressive, and a disturbance around its nest, such as a lawnmower, could lead to several ground bees attacking whoever made the disturbance.
In addition to the aggressive ground bee, there is also the less aggressive mining bee. Instead of the bright yellow and black markings that adorn bees and yellowjackets, these bees are mostly marked by darker colors. Instead of using a pre-existing hole, these bees dig their own holes with entrances usually no larger than a human finger. Mining bees are not as aggressive as their ground bee cousins, but can still sting if they feel threatened.
Whereas worker bees do a variety of chores around the hive and queens repopulate the hive, drones exist only to mate with the female. Unlike the queen and worker bees, which belong to a specific hive, drones wander from hive to hive because they cannot feed themselves. The bee buzzing low around the grass might be a drone, looking for a ground-based hive to enter.