Our evidence suggest that the Romans were pretty big on ghosts, ranging from dead heroes with the status of gods, to the Lares in household ancestor cult, to the generic dead Manes. Unlike modern times, ghosts were not a fringe belief. Ancestor cult played a major role in Roman religion and had a very important place in their lives.

Generally, ghosts are called Manes. A significant number of tombstones start with the letters DM, an abbreviation for dis manibus, consecrating the dead to deified shades. A number of other euphemisms or descriptors are also used - inferi (those below), umbrae (shades), etc. Such spirits were usually considered to be relatively neutral. They generally don't have any effect on the world of the living, but they might show up in dreams. They were propitiated in household cult and also in statewide festivals such as the parentalia (as you might guess from the name, this festival was dedicated to propitiating the ghost of ancestors), the feralia, the rosalia, the Lemuria, etc.

The exact practices for some of these festivals is a little mysterious. Propitiation of ancestors seems to have involved cleaning the tomb, decorating it with flowers (violets and roses seem particularly strongly associated with the dead), offering a sacrifice (usually a libation of wine and a sprinkling of grain mixed with salt, probably also cakes and maybe some other foods), and eating a meal at the graveside with the family. Feralia, Lemuria and some other rites seem to particularly revolve around unrelated/unknown dead who were the type to get particularly testy. These ones were called Lemures (I believe the monkeys are actually named after the ghosts) or Larvae and since they had no family propitiating them were more likely to bring bad luck or sickness, also to be stuck wandering the afterlife without rest or refreshment. They seem to have been propitiated in similar ways to dead ancestors, though there was also a specific nighttime rite for getting rid of lemures which involved black beans (seen to be an appropriate offering to chthonic spirits) and loudly banging pots and pans to scare the ghosts away.

Ovid describes some strange private household rites for the Feralia (Fasti, book II):

This day they call the Feralia because they bear (ferunt)

Offerings to the dead: the last day to propitiate the shades.

See, an old woman sitting amongst the girls performs the rites

Of Tacita, the Silent (though she herself is not silent),

With three fingers, she sets three lumps of incense

Under the sill, where the little mouse makes its secret path:

Then she fastens enchanted threads together with dark lead,

And turns seven black beans over and over in her mouth,

And bakes the head of a sprat in the fire, mouth sewn up

With pitch, pierced right through with a bronze needle.

She drops wine on it too, and she or her friends

Drink the wine that’s left, though she gets most.

On leaving she says: ‘We have sealed up hostile mouths

And unfriendly tongues’: and the old woman exits drunk.

Ovid's account is probably best taken with a grain of salt, partly because he loves him some witches, also he seems to be conflating two different festivals that fell on the same day. Not that that is necessarily wrong, it could be by this period the two had become conflated.

And just for fun, here's one of Marial's descriptions of the underworld and its residents which he wrote to commemorate the death of a slave child and to commend her ghost to the care of his parents, Fronto and Flacilla:

To your shades Fronto, and Flacilla, this child

I commend: she was my sweet and my delight.

Little Erotion shall not fear the darkened shades

nor the vast mouths of the Tartarean hound.

She’d have completed her sixth chill winter,

if she’d not lived a mere six days too few.

Now let her frisk and play among old friends

Now let her chatter, and so lisp my name.

And let the soft turf cover her brittle bones:

earth, lie lightly on her: she lay lightly on you.