prunella vulgaris, Lamiaceae-mint; Heal All
It looks like you have some well developed patches of Prunella. It is a tough, perennial weed.
If you plan to remove it my hand, do it now while the soil is still wet and the weeds relatively easy to pull out. Make sure to dig out all of the roots. You could re-seed the areas this spring.
If you plan to treat the area with an herbicide, it will be most effective to wait until the plants are actively growing later in the spring. Herbicides will be even more effective in the late summer and early fall.
Late summer/early fall is also the ideal time to renovate a lawn. Here is a publication with step by step instructions:
Please let me know if you have any further questions.