I've always been a little confused about this notation. I would really appreciate it if someone could verify that my understanding is correct:
The definition of $M_1 \oplus M_2 \cdots \oplus M_k$ is $M_1 \times M_2 \times \cdots \times M_k$. This is called the external direct sum of $M_1, ..., M_k$
The definition of $M_1+ M_2 + \cdots + M_k$ is $\{m_1 + m_2 + \cdots + m_k|m_i \in M_i \}$. This is called the internal sum of $M_1, ..., M_k$
If it happens that for each $j \in \{1,2, ..., k \}$ we have $M_k \cap (M_1 + \cdots + M_{j-1} + M_{j+1} + \cdots + M_k) = \{ 0 \}$, then $M_1 \oplus \cdots \oplus M_k \cong M_1 + \cdots + M_k$, and abusing notation we write $M_1 \oplus \cdots \oplus M_k$ instead of $M_1 + \cdots + M_k$
Also in the case above, we change the name of $M_1 + \cdots + M_k$ to internal direct sum
If $A, B$ are submodules of $M$ and an exercise asks us to prove that $M = A \oplus B$, then it is actually asking us to prove that $M=A+B$ AND $A \cap B = \{0 \}$, because under the direct product interpretation $M = A \oplus B$ is clearly false.