Creeping Charlie (Glechoma hederacea) is a non-native, shade loving weed that likes poorer soils. In your case, it looks like it has gotten well established in the turf grass.  I cannot tell from the picture if its mainly sunny or shady. Either case, you will need to use herbicide to remove it in the lawn. Best application times are fall and in spring at peak bloom, which, looking at the picture as well as the creeping charlie edging my yard, would be right now for peak bloom time. The fall is an ideal time also, because the grass is dormant but the weed is still green and actively growing.
These sheets from the University's extension service should assist you in best practices.
http://www.extension.umn.edu/garden/diagnose/weed/pdf/weed_list.pdf
http://www.extension.umn.edu/garden/diagnose/weed/idlist.html#CCharlie

If it creeps into your vegetable garden, you will need to hand pull. If hand pulling, care must be taken to get all of the root material, which can be substantial. Do it after or during a rain when the soil is as loose as it is going to get. Loosen the soil around it with your favorite weeding tool. Then grab the plant at its base and wiggle it loose, moving your hand down to the base as far as possible as it is loosening. Remember that if you can't get all of the root material, then this would be a waste of time because the plant will come back. You will need to keep a close eye on the edges of the vegetable garden.   Good luck on eradicating it!