Let $(f_n)$ real function in $L^1$. We say that $f_n\to f$ in distribution if for all $\varphi \in \mathcal C_0^\infty (\mathbb R)$, $$\lim_{n\to \infty }\int_{\mathbb R}f_n\varphi =\int f\varphi .$$
Let $(X_n)$ a sequence of real r.v. We say that $X_n\to X$ in law if $\forall \varphi \in \mathcal C_b(\mathbb R)$ (bounded and continuous functon) $$\lim_{n\to \infty }\int_\Omega \varphi (X_n)\,\mathrm d \mathbb P=\lim_{n\to \infty}\int_{\mathbb R}\varphi (x)f_{X_n}(x)dx=\int_{\mathbb R}\varphi (x)f_X(x)dx=\int_\Omega \varphi (X)d\mathbb P.$$
So in other words, $X_n\to X$ in law $\iff f_{X_n}\to f$ in distribution, except that $\varphi $ is not taken in $\mathcal C_0^\infty (\mathbb R)$ but in $\mathcal C_b(\mathbb R)$.
At the end, are these two notion connected or not really ? Because in one hand, we use test function and on the other hand we use continuous and bounded function.