"Moor" is an anachronistic term that originally referred to people of the Roman province of Mauretania in NW Africa. During the Middle Ages and later, Christian Europeans used the term to refer to Muslims in general, especially the Muslims of al-Andalus (Muslim Spain). Side note: The Spanish applied the term Moor to Muslims they encountered in the Philippines, thus they are called the Moro people.
Regarding the invasion and its causes, the Muslims had conquered all of North Africa by the late 600s, just decades after the beginning of Islam in Arabia. The traditional story is that a Byzantine count, Julian, went to the new rulers of North Africa and asked for an expeditionary force to be dispatched to Iberia to defeat the Visigothic King Roderic, who had supposedly raped Julian's daughter. Tariq ibn Ziyad, the governor of the region around Tangier, responded to this call and cross the Strait of Gibraltar (named after Tariq, from the Arabic Jabal al-Tariq) with a few thousand men in 711. Note: the story of him burning his boats after they crossed is almost certainly false, and was first recorded over 100 years after the crossing itself.
At the Battle of Guadalete on 19 July 711, Tariq's army defeated Roderic (who was suffering from disunity and mutiny among his own officers and soldiers anyways). From that battle, the Visigothic kingdom was basically decimated and Tariq (and his superior Musa ibn Nusayr) spent the next few years conquering the entire peninsula and consolidating it as part of the growing Muslim empire.
The Muslims managed to conquer the entire peninsula save for a small sliver of land up in the northern mountains, where the Kingdom of Asturias remained independent and served as the seed for the Christian conquest in later centuries.
A couple of important notes to keep in mind:
The story of Count Julian and his invitation to Tariq probably never happened. But from the perspective of Islamic law, an offensive war to conquer another region is legal provided there is oppression in that land. In this case, that was probably the case, with the Visigoths (who were originally Germanic) in constant opposition to the general Hispanic population. Religious oppression also existed, with Arian Christians and Jews especially persecuted by the Visigoths. According to Islamic law, however, non-Muslims must be given religious freedoms that were much greater than what was given to non-Christians in Europe at the same time. So a relatively stable and tolerant society developed in Muslim Spain after the conquest.
The fact that Tariq and Musa were able to conquer Iberia relatively easy with only a few thousand men tells of the relative disenchantment of the Hispanic people of Iberia with the Visigothic kingdom. Arabic sources state that many cities voluntarily opened their doors to the Muslim (Moorish) invaders. While that may have been an exaggeration, there didn't seem to be much resistance from the local people.
"Moor" was ironically used as a label for ANY Muslims, including those who were ethnically Hispanic. That's why during the Inquisition in the 1500s, records of the inquisitors regularly state that they were attempting to remove any "Moorish" cultural aspects from society. It wasn't an ethnic identity they were trying to stamp out, it was a religious one.
Sources: The Great Arab Conquests by Hugh Kennedy Tarikh Ibn Khaldun by Ibn Khaldun Lost Islamic History