It depends of the areas inside Spain.
When the Romans came, the people living in Spain were Celtic (half Northwest of Spain) and Iberians (lots of different groups of Iberians: mainly in the East, and South of Spain). So what the Romans saw were what now is called "Celtiberians". That is along with the Romans the basic ethnic sustrate of modern Spaniards. We also had phoenicians, Carthaginians Greeks, suevos, vandalos and alanos (barbarian tribes from Northen Europe), and even vikings in certain villages of Asturias.
After the Romans the ones that ruled Spain were Visigothics which are suppossed that originated in Gothland (what is now Goteborg in Sweden). They ruled Spain for 2 centuries until the 711 ac, time where the Moors invaded Spain (due to a traition between Visighotic kings). The Moors went as far as the North of Spain (even to Potiers in France) during the first 2 centuries but the Spanish kings started to fight against the Moors and pushed them to the South. It was mainly a militar and political invasion (just 30.000 Moors came to Spain in the firsts centuries). The Christian kings liberated the half North of Spain in 2 or 3 centuries and the Moors went to the South. Most of Andalucia was also liberated by the year 1250 and the last Moorish city was Granada which was liberated in 1492.
So the Moors were 8 centuries just in 1 small part of Spain (Granada). In the rest of Spain the time that the Moors were living there was from nothing (in Asturias and other Northen places), 2 centuries in most of the half North of Spain, or 3 or 4 centuries in most of the rest of Spain.
In fact the great majority of the Moors were ethnic Spaniards that converted to Islam to avoid paying the taxes to the Moors rulers. Some studies says that in Granada from a population of 200.000 Moors, just the 1% was a true Moor. The rest were Spanish that converted to Islam.
That is basically the origin of the Spaniards (mainly Iberian (which are suppossed to be what is called "the old Europeans" along with the Basques), Celtic and Romans).
For the curious people: Ireland, Scotland and Welsh celtic people are generally accepted as descendants of the celtic people from Asturias and Galicia. 2 of the 7 celtic countries are Galicia and Asturias. And for the more curious people read about Magnus Maximus who was a Spaniard and is credited to be as the father of the Welsh nation:
http://www.bbc.co.uk/history/bytime/wales/magnus_maximus.shtml "Magnus Maximus was a Roman general of Spanish origin and was active in Britain in c 380 resisting the incursions of the Picts and Scots. In 383, he decided to try to wrest the western Roman Empire from the feeble control of Emperor Gratian. Denuding Britain of much of its garrison, he achieved mastery over Gaul and Spain, but his invasion of Italy led to his defeat and death at Aquilea near Venice in 388. In withdrawing the garrisons, he is believed to have acknowledged the right of some of the Brythonic peoples of Britain to undertake their own defence, an acknowledgement which later historians have seen as a recognition of their right to quasi-independence. This has caused Magnus to be hailed as the father of Brythonic autonomy, and - by extension - the father of the Welsh nation. Furthermore, some of his followers settled in Armorica and thus have been seen as the pioneers of the Brythonic colonisation of Brittany. Indeed, in 1983, festivities were held to commemorate the 1600th anniversary of the foundation of the Breton nation".
or this:
http://homepages.tesco.net/~plk33/plk33/History/FeaturesBritain/BritishMagnusMaximus.htm
It´s a national hero for the Welsh. They changed his name to "Macsen Wledig" as this is how the Welsh know him.