Asymmetric cryptography is a cryptography system which uses public key and private key. Public key can be distributed widely and private key is known only to the owner. The cryptographic algorithm is designed in such a way that it produces one-way function. The distributed public key does not compromise with the security.
So encryption of a message can be done using receiver's public key, but the decryption of that encrypted message can be done only with receiver's private key. This cryptographic system assures confidentiality, robust authentication, data storage etc.
Therefore the most popular use-case of asymmetric cryptography is public key encryption & digital signatures. In case of digital signatures, a message is signed with the sender's private key and can be verified by anyone who has access to the sender's public key. This verification proves that the sender had access to the private key, and therefore is likely to be the person associated with the public key.