Like all species that belong to the Phylum, Ascomycota, Aspergillus can reproduce sexually or asexually. In most cases they reproduce asexually and a few like A. Niger are observed to produce sexually.
Typically, Aspergillus reproduces asexually. Spores are released in the air and then carried by the wind. After landing in a place where there are good conditions, the start to germinate producing foot cells. More branching occurs and elongation of hyphae creating a mass of hyphae or mycelium. Soon conidiophores grow from the foot cells and then the head made of spores (conidial head).
Figure 3.1 Life cycle of Aspergillus (2013 Atsu)
Figure 3.2 Aspergillus nidulans life cycle (Casselton and Zolan 2002 Nature Reviews Genetics)