Nowadays, as a result of the frequent occurrence of accidental injuries and traumas such as bone damage, the number of people causing bone injuries or fractures is increasing around the world. The design and fabrication of ideal bone tissue engineering (BTE) materials have become a research hotspot in the scientific community, and thus provide a novel path for the treatment of bone diseases. Among the materials used to construct scaffolds in BTE, including metals, bioceramics, bioglasses, biomacromolecules, synthetic organic polymers, etc., natural biopolymers have more advantages against them because they can interact with cells well, causing natural polymers to be widely studied and applied in the field of BTE. In particular, alginate has the advantages of excellent biocompatibility, good biodegradability, non-immunogenicity, non-toxicity, wide sources, low price, and easy gelation, enabling itself to be widely used as a biomaterial. However, pure alginate hydrogel as a BTE scaffold material still has many shortcomings, such as insufficient mechanical properties, easy disintegration of materials in physiological environments, and lack of cell-specific recognition sites, which severely limits its clinical application in BTE. In order to overcome the defects of single alginate hydrogels, researchers prepared alginate composite hydrogels by adding one or more materials to the alginate matrix in a certain proportion to improve their bioapplicability. For this reason, this review will introduce in detail the methods for constructing alginate composite hydrogels, including alginate/polymer composite hydrogels, alginate/bioprotein or polypeptide composite hydrogels, alginate/bioceramic composite hydrogels, alginate/bioceramic composite hydrogels, and alginate/nanoclay composite hydrogels, as well as their biological application trends in BTE scaffold materials, and look forward to their future research direction. These alginate composite hydrogel scaffolds exhibit both unexceptionable mechanical and biochemical properties, which exhibit their high application value in bone tissue repair and regeneration, thus providing a theoretical basis for the development and sustainable application of alginate-based functional biomedical materials.