A quick look at: the genius. What was the genius, and how can we view this aspect of Roman domestic religion in ancient art?
A genius (pl. genii) was the divine spirit which the Romans believed every human male was born with; the corresponding guardian spirit in women was called Juno. The genius of the male watched over him throughout his life, and enabled him to beget children. The significance of the genius took on particular importance due to the structure of Roman families.
The Roman family was centered around the paterfamilias, whom was the oldest male member of the family. Everyone within this family was under his control. No major decisions of the family were made without the consent of the paterfamilias, he had control over the property of the family, and for much of Roman history, he had the power of life and death over members of his household. Thus, understandably, the wellbeing of the genius of the paterfamilias was crucial for his entire family, particularly as it was thought to guide the decisions he made. Members of the family would give offerings, and make appeals to the genius of the paterfamilias. Offerings were made on domestic altars (larariums), which nearly every Roman household possessed.
These larariums were usually built in the atrium or kitchen of the home (for an example of a lararium, see this photo from the House of Golden Cupids), and would contain a statuette of the genius (photos 2 & 3). Larariums could also be painted, such as shown in the House of the Vettii at Pompeii (photo 1). Here, we can see the genius figure in the middle, with two lares (household guardian spirits) on either side, to whom offerings were also made. The house snake was also a symbol of the genius, and is often present iconographically in Roman domestic art. These genius figures, be it statuette or painting, are typically depicted as a young, veiled man wearing a toga, whom usually holds a patera and/ or a cornucopia.
The first image is taken by Patricio Lorente via the Wiki Commons, and the shown statuette is courtesy of the Walters Art Museum, Baltimore, USA (54.2329). This figure is made of bronze with silver inlay, and dates to the 1st century.