What is Homomorphic encryption evaluation key In some homomorphic encryption schemes (e.g Multi-key fully homomorphic encryption report, Computing Blindfolded: New Developments in
Fully Homomorphic Encryption), it defines an encryption key, a decryption key, and an evaluation key to perform an evaluation f on the encrypted data. Could you please explain what is the difference between the evaluation key and the other keys?
In short

Public key is used to encrypt, private key is used to decrypt, and evaluation key is used to perform homomorphic operations (usually, homomorphic product or, the somehow equivalent operation, a logic AND gate).

In detail

Public and private keys in homomorphic encryption (HE) schemes are just the same as in other types of schemes.

The evaluation key ($evk$) is also public, it is typically generated using the private key, and it is used to control the noise growth or the ciphertext expansion during homomorphic evaluation.

Some schemes have a "Key-switching" key instead of the evaluation key, but they are more or less the same. For instance, in the description of FV and YASHE, you can see that to perform a homomorphic product, one first multiplies the ciphertexts, $\tilde{c}_{mult} := c_1 \otimes c_2$, then uses this "extra public key" to adjust $\tilde{c}_{mult}$, that is, to get a ciphertext $c_{mult}$ with the correct dimension and that can be decrypted using the original secret key.

So, in general, this is how you use $evk$: you perform a homomorphic operation that introduces a lot of noise or that generates a ciphertext in higher dimension, then you perform an extra operation using $evk$ to "correct" that ciphertext.

As an extra example, take a look to FHEW Library Interface and you will see that the NAND gate uses $evk$, but the NOT gate does not.

Further observations


If you bootstrap, then you typically need $evk$, since you need it to perform homomorphic evaluations in general and bootstrapping is the homomorphic evaluation of the decryption circuit.
In principle, there is no special relation between multi-key HE schemes and evaluation keys (FHEW is not multi-key and also has an $evk$).
Even if the scheme is symmetric (i.e., does not have a public encryption key), it may still have evaluation key.
Some schemes do not have evaluation key (e.g., GSW).