Mouse-ear chickweed. But that's not what's destroying your lawn. Your poor turf is creating the perfect place for that weed to thrive. You need to fix what's causing your lawn to do so poorly. Most post-emergents like Weed-b-gon will knock it down, but you'll need a good pre-emergent application in the fall to keep it under control. It's already flowering and will soon go to seed. Once that happens you will have another generation to deal with.

http://www.weedalert.com/weed_pages/wa_mouseear_chickweed.htm

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John - Zone 6

A renovation may be in order, but I would first start by getting a good soil test done. Hard to say exactly from the two pics, but the turf appears to get enough sunlight so the next thing is to get control of the soil conditions....pH, nutrients, etc. Next step would be to get a good fertilizing and herbicide program going. Another problem may be soil compaction. The best thing you can do to control weeds is to have a thick, healthy turf that is cut regularly at your mower's highest setting.

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John - Zone 6

Not sure, but if they don't, they will surely know where you can get it done locally. Or check with a local county extension agent. They're very inexpensive. I send mine to UCONN. They could do it for you for six bucks if you mail the sample.

http://www.soiltest.uconn.edu/

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John - Zone 6