A description is given of the algorithms implemented in the AstroBEAR adaptive mesh-refinement code for ideal magnetohydrodynamics. The code provides several high-resolution shock-capturing schemes which are constructed to maintain conserved quantities of the flow in a finite-volume sense. Divergence-free magnetic field topologies are maintained to machine precision by collating the components of the magnetic field on a cell-interface staggered grid and utilizing the constrained transport approach for integrating the induction equations. The maintenance of magnetic field topologies on adaptive grids is achieved using prolongation and restriction operators which preserve the divergence and curl of the magnetic field across collocated grids of different resolutions. The robustness and correctness of the code is demonstrated by comparing the numerical solution of various tests with analytical solutions or previously published numerical solutions obtained by other codes.