= Solution
We take algebras to be unital and <modules> to be unital. A finite-dimensional <semisimple algebra> over the <complex numbers> is one whose regular left <module> is a <semisimple module>, that is, a direct sum of <simple modules>. We use the basic finite-dimensional equivalences: this is equivalent to a zero <Jacobson radical>, and the <Jacobson radical> of any finite-dimensional algebra is a nilpotent ideal.
The <Artin–Wedderburn theorem> here says that there are positive integers $d_1,\ldots,d_r$ with
$$
\boxed{A\cong\prod_{i=1}^r M_{d_i}(\mathbb C).}
$$
The nonisomorphic <simple modules> are the standard column <modules> of the factors, of dimensions $d_i$, and the factor sizes are unique up to reordering.
Here is a proof. Decompose the regular left <module> as ${}_AA=\bigoplus_{i=1}^r S_i^{\oplus n_i}$ with pairwise nonisomorphic <simple modules> $S_i$. These are all the <simple modules>: any <simple module> is generated by a nonzero vector and is therefore a quotient of the regular <module>. The <Schur lemma> gives $\operatorname{Hom}_A(S_i,S_j)=0$ for $i\ne j$ and $\operatorname{End}_A(S_i)=\mathbb C$. For the latter assertion, an endomorphism has an <eigenvalue> over $\mathbb C$, and the kernel of its difference from that scalar is a nonzero <submodule>, hence the whole <simple module>. Consequently
$$
\operatorname{End}_A({}_AA)\cong\prod_i M_{n_i}(\mathbb C).
$$
Every regular-module endomorphism is right multiplication by its value at $1$, so $\operatorname{End}_A({}_AA)\cong A^{\mathrm{op}}$. Taking opposite algebras and using matrix transposition gives the asserted decomposition with $d_i=n_i$.
For a <matrix algebra> $M_d(\mathbb C)$, the <matrix units> show that its only <simple module> is $\mathbb C^d$: the spaces $E_{jj}S$ are isomorphic through $E_{ij}$, and a nonzero vector in one of them generates one copy of the column <module>. Simplicity makes that copy all of $S$. In a product algebra, the mutually orthogonal central <idempotents> decompose every <module> into its factor <modules>; a <simple module> uses exactly one factor. This proves the <module> assertion and also uniqueness, since the primitive central <idempotents> and the dimensions of their <simple modules> determine the factors.
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