The ancient Hellenic concept of sacred space can actually be discussed from a variety of perspectives. Burkert, for instance, discusses at great length the slow evolution of different sanctuary styles. He also explains the emphasis on separation of sacred from non-sacred. For instance, a temple might be divided into sections, some of which worshipers would not be allowed to enter (the adyton). Just outside such a sacred space, people would have altars on which to offer sacrifices of various kinds. Even to approach an altar and make sacrifice would require basic purifications and an avoidance of contact with miasma. (A full discussion of miasma and ritual purity is beyond the scope of this article.)
The Perseus Building Catalog is a fantastic on-line resource for images and background information about ancient Hellenic sacred spaces and other works of architecture. A primary source containing a wealth of information is Pausanias's Description of Greece (a translation of which may be found in the Perseus collection).
Altars have been made of a tremendous variety of materials, even some made from the ashes of sacrifices made at a particular site. Although permanent stone altars were not uncommon, Pausanias, in Description of Greece, describes other altars, such as single-use altars made of wood (which would burn down with any sacrifices placed on them):
They fit together quadrangular pieces of wood, putting them together just as if they were making a stone building, and having raised it to a height they place brushwood upon the altar.
Pausanias also recorded altars made of unshaped stones and improvised altars, "their construction being improvised, they are without difficulty taken to pieces".
While some altars are dedicated to a specific deity, others are devoted to several. Beyond just Olympian deities, altars have been dedicated to gods living under the earth, Muses, Sleep, heroes, and other beings. Pausanias even records altars of which different parts are devoted to different purposes (some to deities, but other parts to heroes, nature spirits, rivers, etc.):
The altar shows parts. One part is to Heracles, Zeus, and Apollo Healer, another is given up to heroes and to wives of heroes, the third is to Hestia and Hermes and Amphiaraus and the children of Amphilochus. But Alcmaeon, because of his treatment of Eriphyle, is honored neither in the temple of Amphiaraus nor yet with Amphilochus. The fourth portion of the altar is to Aphrodite and Panacea, and further to Iaso, Health and Athena Healer. The fifth is dedicated to the nymphs and to Pan, and to the rivers Achelous and Cephisus.
In a general sense, altars have been and are used as platforms on which to give sacrifices and pour libations for any of a huge array of reasons. Certain altars may be set aside purely for devotional purposes, whereas others have been tied to oracles, healing miracles, or other special functions. As events surrounding a given altar, deity, or community progress, a given altar's design, use, and purpose may evolve. However, once something is given in sacrifice, it must be given over completely and not reclaimed.
Although in ancient times, ritual slaughter of animals for communal feasting was common, such forms of sacrifice were not appropriate at all times and in all places. Some altars were specifically set aside for offerings of certain types, such as offerings of cakes, coins, etc. Other altars banned certain types of offerings, such as animals and wine.
Libations are discussed in more detail elsewhere on this site, but one traditional way of pouring a libation onto an altar was to pour the beverage from a wide, shallow libation dish not unlike many contemporary serving plates with slightly raised edges.
Burned sacrifices seem to be most common in ancient writings. Barley groats, libations, inedible parts of animals cooked in communal feast, and a wide range of other offerings may be poured, thrown, or placed in the fire to be consumed by the flames.