When I mowed my overgrown UK lawn yesterday, I noticed a network of thin vines. There was also less grass in these patches. I looked and found the leaves of this plant in the areas I hadn't mowed yet. I think I had assumed it was something like clover. What is it? Photos are below.
It's creeping buttercup, Ranunculus repens.
It's highly invasive and quite difficult to eradicate, especially in a lawn. I don't know what country you're in, but you need a lawn weedkiller treatment which acts on this particular weed. In the UK, that would be Verdone Extra; most combined lawn weed and feed treatments won't kill it, you need to use a separate lawn weedkiller. Better lawn maintenance is probably required, although if you live in one of the wetter parts of the UK or the soil is heavy, it won't necessarily get rid of this weed; information on how to manage this generally in lawns here http://www.lawn-craft.co.uk/creepingbuttercup.html
Bamboo is correct...Ranunculus. I have to add that there is a reason your buttercup is very happy. Too much water. I would start training my lawn by watering deeply and not watering again until the grass needs it...
The best way to know when to water is done by walking on the lawn. Stand back to look and if you can still see your foot prints perfectly it is time to water. Water deeply. In this way you are training your grass to grow deep deep roots. This is drought insurance and a water saver! In between waterings your ranunculus and many other lawn weeds are selected out. Too short of roots to get at the water your grass has been trained to reach.
Are you dealing with shade?
What is the percentage off the top of your head Ranunculus to grass? Any other weeds in your lawn? Are you mowing no shorter than 3"...this is a very big deal to stop germination of most weed seeds in a lawn. 3" top growth supports the large root systems of cool season grasses so your grass is able to make enough food for itself to be vigorous and out compete, shade any weed.
Ranunculus is incredibly persistent. I would hold off on herbicide, mow no less than 3", mow once per week, allow that lawn to dry out before watering again! Balanced fertilizer but Ranunculus thrives on that same fertilizer as well.
Ranunculus loves a more acidic soil. The lawn more alkaline. I would also get a soil test or at least a way to measure the pH of your soil. If your pH is less than 6.5 you should lime. Bring it up to 7.0. That will also decrease the vigor of the buttercup.
Train your grass to the point it only needs watering once per week...you might find by footprints you'll need to water maybe twice a week for higher temperatures. In between waterings the buttercup, that loves lots of moisture, dies out. As do most of the shallow rooted weeds. Right now you are selecting for weeds instead of helping your grass compete. Hope this helps?