Let $K$ be a field, $G$ a finite group. Working with linear representation we want to avoid the case where $Char(K)=2$. For this we assume always that $Char(K)\nmid \mid G\mid$. Why we assume this? Can I recover $Char(K)\neq2$ from the second assumption?
Here is my problem, considering the representation on the group $\mathbb{Z}/2\mathbb{Z}$, see linkRepresentation Example
A representation of a group $G$ over a field $K$ is a group homomorphism $\rho \colon G \rightarrow \mathrm{GL}(V)$ for some $K$-vector space $V$. A subrepresentation is a subspace of $V$ on which $G$ acts, i.e. which is closed under the action of the image of $\rho$.
Suppose $G = \mathbb{Z}/2$. Then $G$ is generated by a single element, $g$ say, which satisfies $g^2=e$ (where $e$ is the identity). To define a homomorphism $\rho$ from $G$ to $\mathrm{GL}(V)$ we need to specify $\rho(e)$ and $\rho(g)$. But any such group homomorphism has to send $e$ to the identity $I$ in $\mathrm{GL}(V)$. So we only need to choose the value of $\rho(g)$, which we'll call $\phi$. In order for this to be a homomorphism we must have $\phi^2=I$ (and this is sufficient too).
If $\mathrm{char} K \neq 2$ then any $\phi$ which squares to $I$ is diagonalisable and we can decompose our space $V$ into a direct sum of eigenspaces $V_+$ and $V_-$ of $\phi$, with eigenvalues $+1$ and $-1$. Each subspace $V_+$ and $V_-$ can be written as a direct sum of $1$-dimensional subspaces on which $G$ acts. These are irreducible subrepresentations (i.e. they have no proper subrepresentations themselves).
In summary: if $\mathrm{char} K \neq 2$ then any representation of $\mathbb{Z}/2$ can be written as a direct sum of irreducible subrepresentations; we say the original representation is completely reducible. To see that this can fail if $\mathrm{char} K = 2$, consider $V=K^2$ and $\phi$ given by
$$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}. $$
This squares to $I$ but we cannot decompose $V$ as a direct sum of irreducible subrepresentations.
The correct generalisation of this statement to other groups is that a representation of a finite group $G$ over a field $K$ is completely reducible if $\mathrm{char} K \nmid |G|$. (This is not an 'only if' statement.)