The Khthonians

The Khthonic gods (sometimes spelled chthonic) are those of the Underworld. In the mythos when Zeus and his brothers split the three realms, Haides became King of the Underworld and the Dead (though not of Death, that falls to Thanatos). In some cases the khthonic domains are considered miasmic, such as being around death, though not to the exclusion of their worship. Instead, there is a divide observed between the Ouranic and Khthonic gods and worshiping the two in the same ritual was considered miasmic. Some gods such as Zeus, Dionysos, and Demeter have both Ouranic and Khthonic aspects. 

In the case of Demeter she is of course, one of the regular 12 Olympians and as such Ouranic in that aspect. However, as an agricultural goddess, and literally of the earth, she also has a Khthonic aspect. This is the aspect worshiped in her mysteries which have to do with the cycle of death and rebirth, aspects of the underworld. In praxis this would simply mean worshiping one aspect over the other in a single ritual (i.e. call on Demeter Ourania or Demeter Khthonia). Demeter's worship is a good case study for how the distinction between the two realms were observed traditionally. 

Haides | Ἁιδης

One of the original six siblings who fought in the Titanomachy. He was given the domain to preside over Haides, the Underworld. He is a god of the Dead, though not a god of Death itself (that is reserved for the Titan Thanatos). From the sources surviving he wasn't commonly worshiped. In some cases, his name was considered ominous with the ancient Greeks fearing death would come of invoking the god of the dead. In other cases he was seen as the Zeus of the Dead or Khthonic Zeus, and the god of the dead was the business of the dead. Yet Plutarch says he is kind to Lovers, perhaps due to his happy marriage with Persephone. 

Though he is largely missing from the surviving mythos, there is evidence of temples dedicated to him, and Plato himself suggested dedicating the last month of the year to Haides. In other cases he was even syncretized with Zeus or Dionysos. The relationship that ancient Greece had with the god of the dead is complex, yet in the end he is the god of us all. Death is the natural fate of all humans, not evil or anything to be feared.

Persephone | Περσεφονη

Goddess of Spring, Queen of the Underworld, also known as Kore (typically the name associated with her before her descent into the Underworld, literally means "maiden" and was used to refer to unwed girls). She was worshiped alongside her mother Demeter in the Eleusinian Mysteries which promised a blessed afterlife to initiates. In the mythos she descends into the underworld every year as per her marriage agreement with Haides. Kore's return in Spring is symbolic for her immortality and rebirth. In most other myths she is depicted in her role as Queen of the Underworld.

*Note: Many new worshipers struggle interpreting the myth where she is kidnapped by Haides. While the atrocities of the past should never be excused, if you don't understand the cultural context you will be missing the point of the myth. As far as marriage customs for the time, everything was according to custom and legally binding. Haides had permission from Zeus to wed her, kidnapped her according to bridal custom, and Kore inevitably consented to the marriage by eating the pomegranate seeds. The focus of the myth, then, isn't on the kidnapping itself but rather the grief of Demeter for losing her daughter to the underworld and the founding of their mysteries. This is of course, a very curt explanation.