The word has several meanings: abhāva or absence; pratibandha or obstacle; pīḍā or harassment. However, it is the Nyāya school (of Indian Philosophy) that uses it as a technical term of logic.
In the sentence, parvato vahnimān dhūmāt (‘The hill is fiery, since there is smoke there’), vahni or fire is called ‘sādhya’ (the major term), parvata or hill is called ‘pakṣa’ (the minor term) and dhūma or smoke as ‘hetu’ or ‘liṅga’ (the middle term).
In a situation where the sādhya does not exist in the pakṣa, then the pakṣa is called bādha. For instance, in the sentence hrado vahnimān dhūmāt (‘The lake is fiery, since there is smoke there’) the minor term hrada or lake, which is the pakṣa, is ‘bādha’ since it is impossible for the fire, the sādhya, to exist in the lake. The lake obstructs or prevents (bādha = obstacle) fire from existing in it.
Many more varieties of bādha are discussed in the works of Indian logic.