This weed is pearlwort, a Sagina species. It's either one of the two nearly identical species, S. procumbens or S. decumbens. Whatever their precise name, they are prolific seeders.
Pearlwort prefers cool, moist conditions but will grow year-round. It thrives in moist media/soil of containers, borders, vegetable gardens, lawn, and wherever the soil is exposed.
The tiny seeds are borne in cup-like pods -- as in your image -- and are spread by splashing water. A critical factor in managing the population is to not allow the plants go to seed
The plants are easy to pull up if you gather up the out-spreading stems, the lift straight up. A complication with older plants is that pearlwort roots at each leaf node as the clump enlarges. Thus, the larger the plant, the more roots, the more effort required to remove them, also the more seeds are also scattered for next season's crop.
One resource suggested using a wipe-on treatment with glyphosate, the active ingredient in various RoundUp products. (The directions are on the label.) Painting instead of spraying reduces the risk to nearby desirable plants.
Persistent determination will help you win!