Iberis sempervirens leaf arrangementThis looks like 'Candy Tuff' or Iberis sempervirens. What is amazing is this is not considered a weed. Iberis, great shade ground cover I could be wrong but first impression...wait for others to give their ID's. Never have known this as lawn weed.
Easy to rid your lawn of this guy; mow it. Before it goes to seed. That will probably end this plant in your lawn. Pull it. Keep your grass NO SHORTER THAN 3" this causes enough shade to the soil so that unwanted seeds will have a hard time germinating. Watering deep and allowing the lawn to dry will also kill seedlings or new plants.
No matter the weed's ID this is a broadleaf and there is herbicide that is very selective. Trimec is a common broadleaf weed killer for lawns. Moisture will help the chemicals stick to the BROAD leafs of broad leaf weeds. This chemical does not work for grass weeds.
Too easy to control this plant just by maintenance practices. Send more pictures. How much of this is in your lawn? Is this full sun or partial sun?