= Solution
A <line bundle> $\mathcal L$ is a <globally generated line bundle> when the evaluation map
$$
H^0(X,\mathcal L)\otimes_k\mathcal O_X\longrightarrow\mathcal L
$$
is surjective. Equivalently, its <global sections> span each fibre of the <line bundle>; locally at every point, some section is a generator.
Choose a finite generating family $s_0,\ldots,s_n$. On the open set where $s_i$ generates $\mathcal L$, the ratios $s_j/s_i$ are regular functions, giving a <morphism of schemes> into the standard chart $D_+(x_i)$ of <projective space>. The ratios obey the usual transition rules on overlaps, so these chart maps glue to
$$
\boxed{\varphi:X\longrightarrow\mathbb P^n_k,\qquad x\longmapsto[s_0(x):\cdots:s_n(x)].}
$$
The tuple is computed using any local trivialization of $\mathcal L$; changing that trivialization multiplies all entries by the same invertible function. The construction has $\varphi^*\mathcal O(1)\simeq\mathcal L$ and pulls back the coordinate sections to the chosen $s_i$.
There is a finiteness qualification for an arbitrary $X$: global generation alone need not provide such a finite family. It does if $X$ is <quasi-compact>, since the open sets on which individual sections generate have a finite subcover. Without that hypothesis, take $X=\coprod_{m\geq1}\mathbb P^m_k$ and let $\mathcal L$ restrict to $\mathcal O(1)$ on each component. This <line bundle> is globally generated, but any $N$ global sections have a common zero on a component $\mathbb P^m$ with $m\geq N$. Thus this is a <globally generated line bundle without finite generators>. This is <finite global generation on a quasi-compact scheme>. The finite-family construction is automatic in the projective case asked next.
For projective nonsingular $X$, put $V=\langle s_0,\ldots,s_n\rangle$. \b[The <length-two criterion for a very ample linear system> says that $\varphi$ is a <closed immersion> precisely when, after extending to an algebraic closure, $V$ separates distinct points and tangent directions.] Equivalently, the evaluation
$$
V\longrightarrow H^0(E,\mathcal L|_E)
$$
is surjective for every length-two geometric <closed subscheme> $E\subseteq X$. Two distinct points give point separation; a nonreduced length-two subscheme supported at one point gives separation of a direction in the <Zariski tangent space>. For the complete space $V=H^0(X,\mathcal L)$, this says exactly that $\mathcal L$ is a <very ample line bundle>. For a chosen smaller $V$, it is the chosen <linear system of divisors> which must be a <very ample linear system>; mere global generation is insufficient.
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