We know of the "Tartessian" culture from relatively recent archaeological and linguistic work (mid-20th century to present) in SW Spain, especially at sites like Huelva and Dona Blanca along the corridor of the Guadalquivir River. The culture is typical Early Iron Age, dating (roughly) from the 8th to 5th centuries BCE, though its precursors stretch back into the Iberian Bronze Age. We have a decent number of inscriptions (about one hundred now, I believe), though many do not have a great deal of syntax preserved. As of now, the classification of the language remains unsolved. Many think it is a branch of Celtic, which would make sense, but for now the proof is lacking. Others think it is an isolated, non-Indo-European language with heavy borrowings from Celtic (which could also make sense; cf Basque).

The "Tartessos" name is from Greek, and passed into Greek from the Near East (I think this is the case). It was a legendary place beyond the Pillars of Herculues (Straits of Gibraltar) known for its richness in metals, especially tin and silver. Most of that was simply legend until the middle of the 20th century, when discoveries related to the Phoenician presence in the West were made in Iberia, Sardinia, North Africa, etc by the German Archaeological Service.

We now know that the Phoenicians came to "Tartessos," the region beyond the Straits at the estuary of the Guadalquivir, in the early to mid-8th century BCE, almost certainly in pursuit of metals. They underwent extractive trade arrangements with the Tartessian elite, trading bulk metals for finished goods from Phoenician artisans. Excavations have, in general, uncovered typical Iron Age "Warrior Graves" containing a mix of local Iberian goods and a plethora of Phoenician/Carthaginian, North Syrian, and some Greek finished wares. Around 750-700 BCE, the Phoenicians established an island enclave at Gadir, at the mouth of the river, now called Cadiz. Excavation there is very poor for several reasons, one of which is the presence of the large modern city on top of the site. But up the coast at Dona Blanca, another, smaller Phoenician site at Dona Blanca has enjoyed a modern excavation effort and has produced the types of finds which we now know are typical of Phoenician presence (red slip pottery of predictable and regular type, for example).

The "disappearance" of the Tartessian culture is not really a disappearance. The Phoenician mother cities (Tyre, Sidon) ended their adventure in the West by the middle of the 7th century due to troubles back home in the Levant. Carthage became her own master, but shifted attention to other ventures. Nearly all the Iberian Phoenician sites show signs of decline or abandonment by about 660 BCE (including those at the Guadalquivir and those along the Costa del Sol in Andalusia). The Phoenicians had brought intensive activity and prosperity to the region, and their departure probably only hastened what would have happened anyway: a decline in the archaeologically-visible "warrior graves" as their local Iron Age progressed. After that, Iberia increasingly came under the sphere of Carthaginian influence, and then after that under Roman influence.

There is a very recent book in English about this topic: Sebastián Celestino and Carolina López-Ruiz, Tartessos and the Phoenicians in Iberia (Oxford 2016). I haven't looked at it myself, but I hear it is excellent with the caveat that we (collectively) sometimes want to ascribe too much sophistication to certain societies because they are "ours" (ie in our country). The publisher's summary begins "This is the first book in English about the earliest historical civilization in the western Mediterranean...". No comment.