I will try to answer to the OP by describing at first the relationship between Lie algebras and structure constants. After it, I will move to the problem of integrating a given Lie algebra to a simply connected Lie group. This should help in understanding the
relationship between structure constans and Lie groups.
- On Lie algebras and structure constants
Let $G$ be a Lie group and $\mathfrak g$ be its Lie algebra. We restrict to the finite dimensional case for simplicity. Let $\{e_i\}$ be a basis of $\mathfrak g$ (a basis of infinitesimal generators for $G$): then
$$[e_i,e_j]=f^k_{ij}e_k,$$
where $f^k_{ij}$ are called the structure constants of the Lie group $G$. We are using Einstein's convention on repeated indices. Note that, choosing any other basis $\{v_i\}$ of $\mathfrak g$, we obtain new structure constants
$$[v_i,v_j]=\tilde{f}^k_{ij}v_k,$$
with
$$\tilde{f}^k_{ij}=A_{in}A_{jm}f^s_{nm}A^{-1}_{ks}$$
if $v_i=A_{in}e_n$.
Any Lie algebra $\mathfrak g$ is determined by its structure constants modulo the transformations $f^s_{nm}\mapsto \tilde{f}^s_{nm}$ given above; such transformations determine an equivalence relation. In summary, any two Lie algebras (of the same dimension) with the same structure constants are isomorphic as Lie algebras.
- On Lie algebras and Lie groups
The Lie algebra $\mathfrak g$ captures the local properties of $G$ as it needs only the connected component of the identity in $G$. To answer to your question, it is then necessay to talk about "local vs. global".
On the local level, one has the fundamental theorems of Lie (I do not discuss them here).
On the global level, one needs some work. The theorem below is due to Cartan, and it can be proven using Ado’s theorem, or the Levi-Malcev decomposition. It says that
every Lie algebra of finite dimension over $\mathbb k = \mathbb R, \mathbb C$ is isomorphic to the Lie algebra of a Lie group with some non trivial topological properties. This Lie group is unique up to isomorphisms.
Theorem. If $\mathfrak g$ is a Lie algebra of finite dimension over $\mathbb k = \mathbb R, \mathbb C$, there is a connected simply connected Lie group $G$ such that $\mathfrak g = \operatorname{Lie}(G)$. $G$ is determined uniquely up to isomorphism.
One can translate the above theorem into equivalence of the categories
from the subcategory of connected simply connected Lie groups and the category of finite
dimensional Lie algebras over the base field.
In summary, for any connected simply connected Lie group $G$ one has a unique (up to isomorphisms) finite dimensional Lie algebra $\mathfrak g$ isomorphic to $\operatorname{Lie}(G)$; this Lie algebra is determined by its structure constants as above.
