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Spatiotemporal systems are ubiquitous in a large number of scientific areas, representing underlying knowledge and patterns in the data. Here, a fundamental question usually arises as how to understand and characterize these spatiotemporal systems with a certain data-driven machine learning framework. In this work, we introduce an unsupervised pattern discovery framework, namely, dynamic autoregressive tensor factorization. Our framework is essentially built on the fact that the spatiotemporal systems can be well described by the time-varying autoregression on multivariate or even multidimensional data. In the modeling process, tensor factorization is seamlessly integrated into the time-varying autoregression for discovering spatial and temporal modes/patterns from the spatiotemporal systems in which the spatial factor matrix is assumed to be orthogonal. To evaluate the framework, we apply it to several real-world spatiotemporal datasets, including fluid flow dynamics, international import/export merchandise trade, and urban human mobility. On the international trade dataset with dimensions {country/region, product type, year}, our framework can produce interpretable import/export patterns of countries/regions, while the low-dimensional product patterns are also important for classifying import/export merchandise and understanding systematical differences between import and export. On the ridesharing mobility dataset with dimensions {origin, destination, time}, our framework is helpful for identifying the shift of spatial patterns of urban human mobility that changed between 2019 and 2022. Empirical experiments demonstrate that our framework can discover interpretable and meaningful patterns from the spatiotemporal systems that are both time-varying and multidimensional.