A MAC or authentication tag (sometimes also called signature, although I think that's too confusing) can be used to provide message integrity and message authenticity. Message integrity means that an adversary - or transmission error - cannot change the message without it being detected by the receiver. For physical products we would call this tamper evident. Message authenticity shows that the message originated from a sender holding the secret key. Beware that both sides have the same secret.
Entity authentication is something different: the identification and authentication of parties within a protocol often relies on asymmetric cryptography rather than symmetric cryptography. In the TLS protocol the server is generally authenticated using a private key / certificate pair, while the messages are protected by a authentication tag generated using the symmetric session keys.
Most modes of operation will allow an attacker to make changes for specific bits. For instance, in the popular counter mode (CTR mode) any bit can be flipped, which will lead to a flipped bit in the plaintext at the same position. For CBC mode more bits will be changed, but the decryption and unpadding will still be unaffected (unless the last ciphertext block is changed significantly). One of the other answers underestimates the possibility of locally changing the plaintext by changing the ciphertext.
Generally a MAC is used over the ciphertext instead of over the plaintext. One of the reasons is that some modes of operation - such as CBC with PKCS#7 padding - are vulnerable all by themselves. If a padding or plaintext oracle is possible, CBC may actually allow an attacker to retrieve all the plaintext, without even attacking the cipher or gaining information on the key. For more information, see the question about MAC-then-encrypt and encrypt-then-MAC.
A MAC needs to be over both the ciphertext and the nonce or IV. If the IV or nonce are excluded the plaintext may still be altered by an adversary (if it is accessible by said adversary, of course).
Generally we try and use an authenticated cipher or AEAD cipher. These ciphers will automatically include any nonce or IV within the verification, so they should fully protect the plaintext.
Moreover, they may use a fast MAC that is insecure when used by itself, but is secure if it is combined by the cipher. Examples of these faster ciphers are ChaCha20/Poly1305 and AES-GCM. Both are for instance included in TLS 1.3.
There is even the IP hampered OCB mode which only requires a single pass of the cipher over the plaintext to both encrypt and provide integrity / authenticity. The Keccak sponge (used for SHA-3) may also be used for an authenticated mode.
Note that a MAC in itself doesn't protect against all attacks. For instance, if it is used within a transport protocol, it might be that messages are replayed if the same key is used. Furthermore, since a MAC also relies on symmetric keys, it could be that messages from A to B are captured and send back to A.
It is up to the higher level protocol to guard against such conditions. This is one of the reasons why transport security is hard. Just applying a MAC is not enough for a generic transport protocol. For transport security, use TLS, SSH or another established secure transport layer.