There are at least two ways that the term "liability" may be used. One is a legal term and one is an accounting term.
As a legal term, liability refers to the responsibility that a firm has for the consequences of its actions. For example, if a firm's products cause users to be injured, the firm would have liability for that. In this way, then, liability refers to legal responsibility.
As an accounting term, liabilities are essentially debts. A firm would incur a liability when it borrowed money to expand its plant. It would similarly incur liabilities when it made promises about the pensions that it would pay to retired workers. These are payments that a firm is legally bound to make at some point in the future.