Ground cover in yard
From the leaf shape and leathery texture, I would say that this looks like Gaultheria procumbens otherwise known as Eastern teaberry or American wintergreen. It is native and used as a ground cover because of the evergreen leaves and pretty (often) red berries. (Crush the leaves and you should smell a wintergreen smell). It spreads via underground rhizomes and digging is an option for smaller areas. However, because of the creeping nature some rhizomes may be missed with this process. So the other option (especially for larger areas) is to apply a broad leaf weed killer with the active ingredient of Triclopyr. By using the broad leaf product, grasses and other fine-leaved vegetation will not be affected. If there is no concern about other vegetation, you could also apply a Glyphosate product (such as Roundup) which is non-selective (and will kill all vegetation sprayed). Repeat applications may be necessary and make sure to follow the label directions.
This ground cover may also be a Eunoymous (aka wintercreeper) but most plants in that family spread more like a vine. I can't tell from the picture but if this is a vining plant like Euonymous, you would still use the same products as recommended above.
Hope that helps!