The Aeneid is a great starting place for this, because as a whole the poem is leading up to the eventual founding of Rome - we even get a nice preview of future Roman history in Book 6 - and one of the fun things about Roman domestic religion* is that perhaps more than almost any other area it’s just so particularly Roman. The big deal about the household gods Aeneas brings to Italy is that in some sense they’re used in the poem to signal the difference between Romans and Greeks in religious terms. Romans have them, Greeks don’t.

So who are Vergil’s “household gods”?

Almost certainly in whatever passage you’re reading these are the di penates. The lares, who might also be considered household gods, are only mentioned twice in the Aeneid, only in the singular and only mentioned together with the di penates, and both times the Lar acts as a metonymy for the domestic space or house. This is interesting in and of itself, because it shows the centrality of the Lar to the space of the home, but by contrast a quick concordance search shows 24 results for use of the penates in the poem. I’ll admit I was a bit surprised my search yielded any results for lares at all, as they’re not a particular focus in the epic. So when you see “household gods” in Vergil it’s almost certainly a reference solely to the penates, but the English doesn’t necessarily convey the full valence (we don’t have these categories, so the vocabulary just doesn’t exist): the di penates are household gods, but they’re not the only household gods in ancient Rome.

In a Roman domestic shrine, the three main groups you’ll find in some fashion are the di penates Vergil spends so much time on, together with a pair of Lares, and a genius (the protective spirit of the paterfamilias). The lares and di penates (and sometimes, although not always, a genius) appeared as statuettes in the house's main shrine, giving them a physical presence in the home.

The shrine itself would also be painted, and this allowed for further religious expression. In Pompeii, the typical lararium shrine generally featured a sort of core iconography: the genius, often performing some kind of religious observance, together with the pair of lares and accompanied by two snakes. Other deities could be added to this depiction (Hercules and Mercury were both popular choices in Pompeii), but these formed the main elements that could then be personalized.

Perhaps the best way to differentiate these groups is to imagine the generation-to-generation evolution of a single household shrine. We should start with the lares, because in many ways the lares defined the home. They are gods of place, frequently appearing outside the physical house (there are lots of different types of lares, here I have been referring solely to the lares familiares, the lares of the family) in contexts where boundaries are important. To show some of their importance to the home, brides entering their new house for the first time greeted their lares familiares with an offering before doing anything else. However, one thing lares aren’t is personal. It’s one of those weird things even scholars seem to have a lot of trouble with, given the tendency to conflate Roman religion and Greek mythology, but the lares really don’t have any mythology of the Greek kind. They are iconographically indistinguishable from each other and, except for one prologue speaker in Plautus’ Aulularia, which is tricky to use as evidence, they don’t speak. The presence of a Lar defines the lifespan of a domestic shrine, but they have no individualized identity.

Next up the di penates were a family’s gods, and they could be added to over time as the family decided to adopt them. The di penates are also the household shrine’s connection to what we might think of as more traditional larger scale cult (to gods whose names you would recognize like Juno or Vesta). These groupings were individual, so they could also include local gods not typically part of Roman worship and could include deities far beyond the conventional Olympian set. These came from a combination of family inheritance, travel, local cult, and personal experience. These were not generally removed once introduced, and we have some examples where multiple statues of the same god appear in the same assemblage, but the existence of any particular one of the di penates wasn’t necessary, so house to house any one god might appear and disappear, and secondary shrines in the house never included them at all unless they were painted into the lararium itself. This is also what answers your question about the material nature of the penates. These deities were personal, so keeping safe the statue of your Juno was important, since your Juno is not the same as the cult to Juno at the temple. In fact, Cicero makes quite a big deal of saving his personal Minerva when he is sent into exile, apparently relocating her to a temple of Jupiter while he is gone. So to recap, while the lares are more de-individualized, the di penates are personal, family-specific gods passed down over generations.

The genius, by contrast, is tied to the lifetime of an individual. He is a god, representing the whole household, but since he is associated with the paterfamilias, the genius was thought to appear and disappear with the individual birth and death. On rare occasions, we have also found iconographic representation of the Juno, the equivalent for the wife of the paterfamilias. The role of the paterfamillias in the cult of the lares is also why in paintings the genius is usually depicted in the process of ritual action, making an offering or libation to the lares.

These deities together are the inhabitants of a Roman domestic shrine, the household gods. In different ways, whether because of their centrality to the very essence of the home or their personal connection, they are all very much part of present life in the family and so their place in the house makes sense in that light. All of these elements are also particularly Roman. The big deal in the Aeneid about Aeneas bringing the penates to Italy isn’t just that it’s an act of pietas, although it is, it’s that Vergil is attempting to give a fundamentally Roman practice an origin within the Homeric tradition while also marking out Aeneas and his descendants as being different from the Greeks.

As an (hopefully brief) addendum, since I realize I never touched upon ancestor worship: the short answer is that the dead, the di manes, had no place in the home, although they did receive - separate - worship. The slightly longer answer is that the antiquarian tradition conflated the lares and the di manes, and it’s led to confusion ever since. There are plenty of possible culprits for the modern scholarly conflation of the two, but that’s really an issue of modern approaches. No matter the reason, the lares and the di manes remain separate groups with their own distinct spheres.

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*I use the term “domestic religion” throughout here as a way of grouping together religious activity that happened in the home, but as a concept “domestic religion” in the sense of private religion has faced criticism for the false dichotomy it presents between public and private religious life, as if there exists some sort of hard boundary between the two. Acknowledging this, I’m going to go ahead and use it anyway because in this instance it’s just convenient. We’re talking about religion in the house, we’re going to call it domestic.

Bodel, John. “Cicero’s Minerva, Penates, and the Mother of the Lares: An Outline of Roman Domestic Religion.” In Household and Family Religion in Antiquity. Edited by John Bodel and Saul M. Olyan. Blackwell, 2009. 248-275.

Flower, Harriet. The Dancing Lares and the Serpent in the Garden: Religion at the Roman Street Corner. Princeton: Princeton University Press, 2017.

Kaufmann-Heinimann, Annemarie. “Religion in the House.” In A Companion to Roman Religion. Edited by Jörg Rüpke. Blackwell, 2007. 188-201.

Rüpke, Jörg. Pantheon: A New History of Roman Religion. Princeton: Princeton University Press, 2018.

It's not a direct answer, so there is certainly more to be said, but /u/tinyblondeduckling addressed a lot of issues around the household gods in an answer to How did Roman families acquire Lares?