A typical example arising in algebraic geometry is the following.
Take a scheme $X$ of finite type over $\mathbb{C}$. Then there are (at least) two concepts of fundamental group for $X$: the topological fundamental group $\pi_1^{top}(X)$, i.e. the fundamental group of the underlying topological space $X(\mathbb{C})$ with the analytic topology, and the étale fundamental group $\pi^{et}(X)$, which is a purely algebraic object (it is an inverse limit of finite automorphism groups).
By the so-called Riemann Existence Theorem, the finite covering of $X(\mathbb{C})$, that a priori are only analytic spaces, have actually a scheme structure. This implies that $\pi_1^{et}(X)$ is precisely the profinite completion of $\pi_1^{top}(X)$.
Therefore there is a group homomorphism $$\eta \colon \pi_1^{top}(X) \longrightarrow \pi_1^{et}(X),$$ which is injective precisely when $X$ is residually finite.
For instance, if for some reason we know that $\pi_1^{top}(X)$ is residually finite and we are able to show that $\pi_1^{et}(X)=0,$ then we can conclude that $\pi_1^{top}(X)=0.$
J. P. Serre asked if there exist smooth, complex varieties such that $\pi_1^{top}(X)$ is not residually finite (and hence $\eta$ is not injective). A consequence of a theorem of Malcev is that if $\pi_1^{top}(X)$ has a faithful linear representation, then it is residually finite. Hence a variety answering affirmatively Serre's question must have the topological fundamental group without any faithful linear representation. Such varieties actually exist, and the first examples were constructed by D. Toledo in 1993, see
Projective varieties with non-residually finite fundamental group, Publications Mathématiques de l'Institut des Hautes Études Scientifiques Volume 77, Issue 1 (1993), 103-119.