There are two types of liability coverages: "occurrence" and "claims made." Most liability insurance is written on what insurers call an occurrence basis. With this type of coverage, you are contractually obligated to notify the insurer as soon as you become aware that an event (the occurrence) has caused, or seems likely to result in, bodily injury, or property damage for which you may be legally liable. That insurer stays with the claim until it is resolved, however long that may be. Even if you later buy liability coverage from a different insurer, the insurer to whom you originally reported the occurrence stays with the claim.
With claims made liability coverage, your current insurer covers all the claims you make during the policy's coverage period, even those claims due to injury or damage that is the result of an occurrence prior to the current policy period. This type of policy is most prevalent in areas such as Professional Liability Insurance, where there can be many years between an occurrence and the recognition of damage or bodily injury. It could, for example, be years after an architect designs a building before part of the building collapses and the owner sues the architect claiming that the collapse was the result of errors in the design. With a "claims made" policy, the insurer that insures the architect for professional liability at the time the claim is made covers the claim.