Home compost means that you have a place at home, likely a DIY compost bin, where you can place your organic waste for composting. Typically, home compost is used for gardening purposes.
Community composting involves a collecting organization that will take your organic waste from your home compost bin to a place where a large compost pile will be assembled and used afterwards to regenerate and fertilize soils. Consult with your local composting or recycling organization to see which organic materials they accept in their community curbside or drop-off composting programs. A list of common composting items can be found here.