Owing to abundant spectrum resources, millimeter wave (mmwave) communication promises to provide Gbps data rates, which, however, may be restricted by large path-loss. Thus, antenna arrays are commonly used along with beam alignment (BA) as an important step to achieve the array gain. Efficient BA relies on the beam training codebook design. In this paper, we propose a new hierarchical codebook to achieve uniform BA performance with low overhead. To better elaborate on the design principle, a single-path channel model is considered first to frame the proposal. The codebook design is formulated as an optimization problem, where the ripple in the main/side lobes is constrained such that each training beam is close to the ideal one with a flat magnitude response and a narrow transition band. Then, we propose an efficient algorithm to find such a beam training codebook. Furthermore, we derive closed-form expressions of the BA misalignment probability or error rate of the proposed beam training codebook. Our results reveal that using the proposed codebook, the error rate of tree-search-based BA exponentially decreases with the SNR for a given channel, and linearly decreases in the log-log coordinate axis for a fading channel. We further propose a power allocation scheme used in different training stages to further improve the BA performance. Finally, the proposed framework is extended to the more complex case of multi-path channels. Numerical results confirm the effectiveness of the proposed training codebook and power allocation scheme as well as the accuracy of the performance analysis.