This is Pearlwort, it is a troublesome perennial weed in lawns, it can be mistaken for moss. Unfortunately it is able to grow and thrive in regularly mown lawns. It germinates and spreads quickly so early treatment is necessary to eradicate it. The presence of weeds in the lawn usually indicates that grass is not growing sufficiently vigorously to prevent weeds from establishing so it may be a good idea to start treating your lawn and giving it some tlc. Hand weeding is effective when removing a few scattered weeds but a selective herbicide (singular or a weed and feed type) is probably the answer here. The lawn will probably require a few applications due to the persistence of seeds to germinate and spread.
Good luck with this problem.
Lynn Burton - Horticultural Adviser