Short answers, pulled from the story.
The term Nationalists was coined by Joseph Goebbels after a clandestine Spanish delegation led by Captain Francisco Arranz visited Germany on the 24th of July 1936 to request war materiel. Goebbels chose the label to lend legitimacy to Nazi Germany's support for the Spanish rebel military.
The Nationalist faction included the Falange, the Catholic right-wing CEDA, and two rival monarchist movements: the Alfonsist Renovación Española and the Carlist Traditionalist Communion. In 1937 Franco merged all these groups into the Falange Española Tradicionalista y de las JONS under his personal leadership.
Nazi Germany, Fascist Italy, and Portugal all supported the Nationalists. Germany sent the Condor Legion along with tanks and aircraft including the Panzer I, Messerschmitt Bf 109, and Heinkel He 111. Italy deployed 35,000 troops in the Corpo Truppe Volontarie by January 1937. Portugal provided ammunition, logistics, and an unofficial volunteer force of between 8,000 and 12,000 fighters known as the Viriatos.
The Army of Africa was a field army garrisoned in Spanish Morocco, comprising the Spanish Foreign Legion and the Regulares recruited from the Moroccan population. More than 13,000 Moroccan troops were airlifted to mainland Spain on 20 Junkers Ju 52 planes between July and October 1936. It became the most decorated unit in the May 1939 Nationalist victory parade, with an estimated one in five of its members killed during the war.
Franco moved to consolidate power after the deaths of the faction's early leaders. In 1937 he issued a unification decree folding the Falange and the Carlists into a single movement, the Falange Española Tradicionalista y de las JONS, with himself as leader. He also outmaneuvered Manuel Hedilla, who had sought to lead the Falange after José Antonio Primo de Rivera's death. Franco then distanced the unified party from fascist labels and promoted National Catholicism in place of the Falange's original anticlerical stance.
The Catholic Church formally sided with the Nationalist rebels, describing Spanish fighters killed by Republican forces as martyrs of the faith and supporting the framing of the conflict as a religious Crusade. The Holy See used its diplomatic network to lobby for the rebel side and by 1938 was one of the first states to officially recognize Franco's Spanish State. At a 1937 International Art Exhibition in Paris, the Vatican allowed the Nationalist pavilion to display under the Vatican flag because Franco's regime was not yet formally recognized.