What is this weed and how do I get it to stop taking over my lawn?
The weed looks like it may be a type of pennywort or dollarweed, Hydrocotyle spp.
It is a warm season perennial weed that spreads by seeds, rhizomes, and tubers. Dollarweed thrives in weak, thin turf with excessive moisture. The first defense against dollarweed is to reduce moisture levels and modify cultural methods (i.e., proper mowing height and irrigation).
Here is some information from Clemson on this weed https://hgic.clemson.edu/factsheet/dollarweed/
You will have to make some decisions. You can live with the weed and practice good cultural practices listed above. Here is our publication on Turfgrass Maintenance throughout the season. http://extension.umd.edu/sites/extension.umd.edu/files/_docs/programs/hgic/HGIC_Pubs/lawn_pubs/HG112...
If you want to use chemical control, you will have to consider total lawn renovation.
as it looks like the weed has overtaken your entire lawn more than 40-50%.
This includes having your soil tested http://extension.umd.edu/hgic/soils/soil-testing, decide on seed, kill the existing lawn with glyphosate, prepare the site for planting, and seed or install sod. Now is the time for lawn renovation for a cool season grass such as tall fescue. Tall fescue grows best in full sun to partial shade. Fine fescue is recommended if the lawn is mostlyi shade. Hard fescue is the best choice.
Here is our publication on Lawn Establishment and Renovation see page 7 and our website http://extension.umd.edu/hgic/lawns/lawn-renovation
Here is more information from our IPM Report page 3 https://extension.umd.edu/sites/extension.umd.edu/files/_docs/programs/ipmnet/13Oct11L.pdf
You can use a post emergent herbicide next growing season as soon as you notice the weed to reduce any further growth. You can use a combination herbicide that contains the following broadleaf weed killers: 2,4- D, dicamba, and mecoprop (MCPP) or MCPA. Follow label directions as caution needs to be used around the roots of trees and shrubs.