What is this?
This looks like Houttuynia cordata, chameleon plant. This is a non-native invasive plant that spreads by its rhizomes (roots). Kill it as fast as possible. We get many requests for help to get rid of this plant--which is still being sold by the nursery industry.
Usually, it is sold in a variegated version known as 'Chameleon Plant.' But, after a few years, it reverts to its original color. Either way, it rapidly becomes obnoxious and nearly impossible to get rid of.
Due to its aggressive nature it is extremely difficult to eliminate. If you have no other plants in the garden bed, you can try hitting it with glyphosate, a nonselective systemic herbicide (that kills the roots) and then hand dig the plants that the chemical does not kill. Glyphosate is found in products such as Roundup and will kill all types of plant material, so you must be very careful to not apply it to the lawn or other desirable plants. It will probably take several applications to have any noticeable effect on the Houttuynia. When you have killed off some the plants then dig the remaining plants. This plant spreads by rhizomes. You must be sure to dig up all of these underground horizontal stems or they will sprout new growth. You will then need to monitor the area for new growth and spray or dig as it appears. It will probably take more than a year to rid the area of the Houttuynia . Do not replant the area until you are sure all of the Houttuynia is gone. Consider this a military campaign, not a single battle.
If the Houttuynia is mixed in a bed with other plants, it is riskier to use chemicals because you may kill the desirable plants. You have a few options. You can hand dig the Houttuynia. Again it will take some time to get all of the plants and their rhizomes. You could try painting the glyphosate on the Houttuynia with a small foam paint brush. Or spray, but use a shield of cardboard or plastic to keep spray off desirable plants. By doing this you would be less likely to apply the chemical on the desired plants. Then hand dig any Houttuynia that is not killed by the chemical.
A last option is to remove the desirable plants to another bed and use the chemical and mechanical method described in the above paragraph. If you decide to do the last option, be very careful not to move any of the Houttuynia rhizomes to the new bed. You may want to wash or shake almost all the soil off the plant roots to be sure no houttuynia is entangled. These rhizomes could very easily be mixed with the desirable plant roots.
Here are links to a couple of sites that discuss Houttuynia removal.
Wow! Thank you for your detailed response. Looks like I have a lot of work ahead of me. :)