Carbon dioxide is intimately associated with metabolism in plants and animals - it is a building block and a waste product. Like water, it is not “organic” and yet there would be no life on Earth without it. So harnessing the processes of biology that consume CO2 as a building block have allure as ways of removing it from the environment; any processes that create biomass are worth examining - forestry practices, agriculture, production of both microalgae and macroalgae, and enhancements to ecosystems in coastal marine zones and soils are all possibilities. All these processes ultimately rely on photosynthesis to convert carbon dioxide into glucose, which green plants can use for energy or to make cellulose and other structural chemicals.
Because carbon dioxide is not organic and does not decompose, it is inherently a stable form of carbon. Chemically, it is a short path away from becoming carbonate - which is often a component of rocky materials including vast portions of the Earth’s crust. Carbon dioxide is always dissolving into water from the atmosphere, and there are opportunities to accelerate that phenomenon. And there are many chemicals which have a greater affinity for carbon dioxide than even water. Processes that harness these processes of dissolution and adsorption in engineered systems are often known as Direct Air Capture, though the conversion of carbon dioxide to carbonate can also occur as part of the carbonate-silicate geological cycle which we are learning to accelerate in various ways; enhanced rock weathering, ocean alkalinization, and geological sequestration.
Biologically captured carbon dioxide is converted to biomass which will eventually decompose. The decomposition process will return the carbon to the atmosphere. So by themselves, biological methods of CDR have less permanence. Physical and chemical processes applied to biomass can increase the permanence of carbon storage by either preventing or delaying decomposition, or by converting biomass into non-biodegradable forms - BECCS, biochar, and production of engineered materials are some of the ways of storing the carbon in biomass.