No. A moss has nothing a botanist would call a true root. What looks like one, the tangle of fine brown threads at the base of a cushion, is a set of rhizoids, and their work is to hold the plant in place rather than to feed it. That single fact sits behind almost everything odd about how moss lives and how you look after it.
What rhizoids actually do
Peel a clump of moss gently off a wall and you will see them: pale to rusty filaments, finer than a hair, matting the underside of the cushion and gripping whatever it grew on. Each is only a cell or two wide, often with slanting cross-walls, and it branches as it works into every pit and crevice of the surface. Some mosses knit them into a dense brown felt, a tomentum, that clothes the lower stem; others carry only a sparse fringe. The job is mechanical either way. Rhizoids anchor, and they do not draw water or minerals up into the plant the way a root feeds a flower or a tree. A moss will grow perfectly well with almost none of them, so long as something else holds it steady.
So where does its food come from?
Straight off its own surface. A moss has no vessels inside it, no xylem or phloem running up a stem, so it cannot pump anything from base to tip. Each leaf, usually a single layer of cells with no waterproof skin, drinks directly from rain, mist and dew as they settle on it. The minerals the plant needs arrive dissolved in that same water, together with the fine dust that falls out of the air, so a cushion is nourished evenly all over rather than from one end. Plants that take water in across the whole shoot like this are called ectohydric, and it is why a moss can thrive on bare rock or a roof tile that offers it nothing but a foothold.
Why the grip still matters
None of that makes rhizoids trivial. A moss that cannot hold on is a moss washed into the gutter by the first heavy shower, so anchorage decides whether a patch establishes at all. The felt at the base of a cushion does a quieter job besides, trapping a film of moisture and keeping the humid stillness the plant likes around its lower parts. On loose or crumbling ground some mosses bind the surface together as they take hold, which is part of how bare soil is steadied for the plants that come after. Grip is not a poor imitation of a root system; it is the one part of a root's work that a moss genuinely needs.
What it means when you grow it
Everything awkward about establishing moss follows from this rootlessness. You do not plant a moss, you lay it on, pressing it into firm contact so the young rhizoids can catch; bury the green stems in soil as you would a seedling and you simply smother them. Feeding happens through the water and the air, never through the ground, so there is little to gain from enriching the soil under a moss lawn and a great deal to gain from keeping the surface damp and clean. Because the plant asks only for something to cling to, it will settle on brick, stone, bark, concrete and trodden earth alike, and it spreads readily from mere fragments, each broken shoot able to grow fresh rhizoids and begin again. The companion pages on growing moss and water for moss both lean on exactly these habits.
The mossy plants that do have roots
One caution is worth adding, because the names mislead. Clubmosses and spikemosses, the wiry little Lycopodium and Selaginella plants sold for terrariums, look mossy but belong to a separate, vascular lineage with real roots and internal plumbing. The true mosses, the soft leafy cushions and carpets this site is about, never do. If a green mat is hauling itself up on genuine roots, it is not a moss at all; sorting the real thing from its imitators is the subject of telling moss apart from its lookalikes. For the rest of how the whole plant is built, the biology of mosses fills in the picture.