How does the Efficient Fully Homomorphic Encryption from (Standard) LWE work?
The BV scheme presented in the article works for plaintext messages in $GF(2)$ (that is $m$ $\in$ $\{0,1\}$) but can be extended to support encrypting messages that are in $GF(t)$ where $t$ is prime with the modulus $Z_q$.

This means that if you want to encrypt a message in $\{0,1\}^n$ you have to do it for each bit separately in $GF(2)$ (or in $GF(t)$ and the message $m$ $\in \{0,1,...t-1\}^n$)

The BGV scheme instantiated with the LWE problem is the same as in BV.
The main difference in BGV is that the scheme supports a more general context, namely it could be also instantiated with the Ring-LWE problem providing a more efficient circuit (inner operations are done inside a ring $R$ instead from $Z_q$ as in LWE).