Malleable encryption
Several cryptosystems possess this partially homomorphic property. Notable examples include Benaloh, and Naccache-Stern which generalizes it, as well as Damgård–Jurik which generalizes the Paillier cryptosystem. 

A worked example of the latter scheme:

The encryption primitive is defined as $E(m)=g^m\cdot r^n \mod n^2$ for a random element $r \in \mathbb{Z}$. From this we can see that given two ciphertexts we have:
$$E(m_0)\cdot E(m_1) = (g^{m_0}\cdot {r_0}^n) \cdot (g^{m_1}\cdot {r_1}^n) \mod n^2$$
$$E(m_0)\cdot E(m_1) = g^{m_0 + m_1}\cdot {(r_0 \cdot r_1)}^n \mod n^2$$
$$E(m_0)\cdot E(m_1) = E(m_0 + m_1) \mod n^2$$

So we can compute the encryption of the addition of two plaintexts from only the ciphertexts, without revealing them, by simply multiplying the ciphertexts.

Additionally, if we know the value of $m_1$ then we can avoid using the encryption primitive and directly compute as follows:
$$E(m_0)\cdot g^{m_1} = (g^{m_0}\cdot {r_0}^n) \cdot g^{m_1} \mod n^2$$
$$E(m_0)\cdot g^{m_1} = g^{m_0 + m_1}\cdot {r_0}^n \mod n^2$$
$$E(m_0)\cdot g^{m_1} = E(m_0 + m_1) \mod n^2$$

Now, by viewing multiplication as repeated addition, we can extend the homomorphic property to multiply by a constant, by implementing repeated ciphertext multiplication using exponentiation:
$$E(m_0)^{m_1} = (g^{m_0}\cdot {r_0}^n)^{m_1} \mod n^2$$
$$E(m_0)^{m_1} = g^{m_0 \cdot m_1}\cdot {({r_0}^{m_1})}^n \mod n^2$$
$$E(m_0)^{m_1} = E(m_0 \cdot m_1) \mod n^2$$

EDIT: Thanks tylo for specific examples