Short answer: On-exchange plans are bought through the marketplace (HealthCare.gov or a state exchange) and are the only way to get premium tax credits. Off-exchange plans are bought directly from an insurer or broker; they follow the same ACA rules but are not eligible for subsidies.
Individual ACA-compliant plans come in two channels. On-exchange (marketplace) plans are purchased through HealthCare.gov or a state-run marketplace. Off-exchange plans are the same type of ACA-compliant coverage but bought directly from an insurer or through a broker, outside the marketplace.
The decisive difference is financial help. Premium tax credits and cost-sharing reductions are only available for on-exchange plans, so anyone who qualifies for a subsidy should shop on the marketplace. Off-exchange plans never carry subsidies, but insurers sometimes offer plan designs or networks off-exchange that are not listed on the marketplace, which can appeal to people who do not qualify for help.
If your income may make you eligible for premium tax credits, start on the exchange. If you are confident you will not qualify for a subsidy, it is worth comparing both channels, since the same carrier may offer different options on and off the marketplace. Both must cover essential health benefits and cannot deny coverage for pre-existing conditions.