WHAT ARE EXTRATERRESTRIAL ARTEFACTS?
Conventional archaeology studies human artefacts on the Earth. However, there are now human artefacts on other celestial bodies (eg the Moon and Mars). These might be referred to as extra-terrestrial artefacts.
Another class of extraterrestrial artefact might be artefacts left behind as a result of an alien culture that had either visited the Earth or a body near the Earth, or that had sent a probe to Earth or the space near the Earth. Even if the artefact was found on the Earth it would probably be called an extraterrestrial artefact because it was the result of activities of an extraterrestrial culture.
HOW TO RECOGNISE AN ETA

WHERE TO SEARCH
LUNAR SEARCHES
The image below was taken by the Lunar Reconnaissance Orbiter of the Apollo 11 site, showing artefacts left behind by the astronauts.

Some people have claimed that ET left an image of a 'face' on Mars, on the basis of a Viking 2 Orbiter image of the Cydonia region of the planet.

Carl Sagan gave good advice in trying to evaluate such claims:
"Extraordinary claims require extraordinary evidence."
And indeed a higher resolution image and images taken at different angles do not show a face.
It has also been said that ETs are currently present on the Earth. This claim is addressed in the article on UFOs.
REFERENCES
Robert A Freitas & Francisco Valdes, “The Search for Extraterrestrial Artifacts”, Acta Astronautica, 12, 1027-1034 (1985)
Paul Davies & RV Wagner, “Searching for Alien Artifacts on the Moon”, Acta Astronautica, 89, 261-265 (2013)
FICTION
Arthur C Clarke, The Sentinel, (short story) Avon Periodicals 1951.
Australian Space Academy