Decompositions of tensors into factor matrices, which interact through a core tensor, have found numerous applications in signal processing and machine learning. A more general tensor model that represents data as an ordered network of subtensors of order-2 or order-3 has, so far, not been widely considered in these fields, although this so-called tensor network (TN) decomposition has been long studied in quantum physics and scientific computing. In this article, we present novel algorithms and applications of TN decompositions, with a particular focus on the tensor train (TT) decomposition and its variants. The novel algorithms developed for the TT decomposition update, in an alternating way, one or several core tensors at each iteration and exhibit enhanced mathematical tractability and scalability for large-scale data tensors. For rigor, the cases of the given ranks, given approximation error, and the given error bound are all considered. The proposed algorithms provide well-balanced TT-decompositions and are tested in the classic paradigms of blind source separation from a single mixture, denoising, and feature extraction, achieving superior performance over the widely used truncated algorithms for TT decomposition.