The crystalline structure, magnetism, and magnetocaloric effect of a GdCrO$_3$ single crystal grown with the laser-diode-heated floating-zone technique have been studied. The GdCrO$_3$ single crystal crystallizes into an orthorhombic structure with the space group $Pmnb$ at room temperature. Upon cooling, under a magnetic field of 0.1 T, it undergoes a magnetic phase transition at $T_{\textrm{N-Cr}} =$ 169.28(2) K with Cr$^{3+}$ ions forming a canted antiferromagnetic (AFM) structure, accompanied by a weak ferromagnetism. Subsequently, a spin reorientation takes place at $T_{\textrm{SR}} =$ 5.18(2) K due to Gd$^{3+}$-Cr$^{3+}$ magnetic couplings. Finally, the long-range AFM order of Gd$^{3+}$ ions establishes at $T_{\textrm{N-Gd}} =$ 2.10(2) K. Taking into account the temperature-(in)dependent components of Cr$^{3+}$ moments, we obtained an ideal model in describing the paramagnetic behavior of Gd$^{3+}$ ions within 30--140 K. We observed a magnetic reversal (positive $\rightarrow$ negative $\rightarrow$ positive) at 50 Oe with a minimum centering around 162 K. In the studied temperature range of 1.8--300 K, there exists a strong competition between magnetic susceptibilities of Gd$^{3+}$ and Cr$^{3+}$ ions, leading to puzzling magnetic phenomena. We have built the magnetic-field-dependent phase diagrams of $T_{\textrm{N-Gd}}$, $T_{\textrm{SR}}$, and $T_{\textrm{N-Cr}}$, shedding light on the nature of the intriguing magnetism. Moreover, we calculated the magnetic entropy change and obtained a maximum value at 6 K and ${\Delta}{\mu}_0H$ = 14 T, i.e., --${\Delta}S_{\textrm{M}} \approx$ 57.5 J/kg.K. Among all RECrO$_3$ (RE = $4f^n$ rare earths, $n =$ 7--14) compounds, the single-crystal GdCrO$_3$ compound exhibits the highest magnetic entropy change, as well as an enhanced adiabatic temperature, casting a prominent magnetocaloric effect for potential application in magnetic refrigeration.