Risk-attaching basis -
A basis under which reinsurance is provided for claims arising from policies commencing during the period to which the reinsurance relates. The insurer knows there is coverage for the whole policy period when written.
All claims from cedant underlying policies incepting during the period of the reinsurance contract are covered even if they occur after the expiration date of the reinsurance contract. Any claims from cedant underlying policies incepting outside the period of the reinsurance contract are not covered even if they occur during the period of the reinsurance contract.

Loss-occurring basis -
A Reinsurance treaty from under which all claims occurring during the period of the contract, irrespective of when the underlying policies incepted, are covered. Any claims occurring after the contract expiration date are not covered.
As opposed to claims-made policy. Insurance coverage is provided for losses occurring in the defined period. This is the usual basis of cover for most policies.

Claims-made basis -
A policy which covers all claims reported to an insurer within the policy period irrespective of when they occurred

Few more terms on reinsurance from wikipedia.org