Spain never entered the Second World War for a variety of reasons, the biggest one being the fact that the civil war had only ended 6 months before the onset of WW2. This left Spain a shell of its pre war state with over 500,000 killed and another half a million having left the country (to put that into perspective, Spain in 1936 had about 25 million people). The war weariness of the population after those three years plus immense industrial destruction - Spain never even had an huge industry before the civil war, lagging severely behind every comparable European country by a large margin - meant Spain wasn't really in a position to wage war on a global scale.

However, there were talks between Hitler and Franco, culminating in a meeting at Hendaye in October, 1940, over Spain's entry into the war as an ally of Germany. Franco's ambitions were to take part of the French colonial empire, namely parts of Morocco and north Africa as France collapsed under the German invasion, and to enter an alliance to counter British hegemony in the region by seizing Gibraltar. The Germans, however, rejected Franco's demands, believing them to be too ambitious and that Spain would be of little use as an active combatant ally in the war (as said before, Spain was in a sorry state). In addition, Hitler determined that keeping Vichy France on side was more important than gaining Spain (Spanish demands included what was, following French collapse, part of the Vichy colonial territory, which would not have pleased the Vichy government). Despite the rejections, Franco did however allow German intelligence networks to operate in Spain and German ships were allowed to use Spanish ports.

So that is why Spain didn't join the axis during the war, but why was Spain tolerated after? The Republicans in exile were hoping that, after liberating France, the Allies would move to liberate Spain too from the axis aligned regime. Republican exiles in France, who had been fighting in the resistance against German occupation, organised an assault over the border in the Aran Valley, in the hope that this would spark revolutions within Spain which would then be supported by advancing Allied troops. However, this didn't happen. After briefly holding a few towns, the Spanish Army quickly repelled them, and Allied support never came. For the Allies, Spain was not a high priority and not worth the risk - the Pyrenees especially were a formidable defence, even against the barely recovering Spanish Army, and Spain was being cooperative in leaving Gibraltar in peace.

After the conclusion of the war, after a period of isolation lasting until the early-mid 1950s, Franco started to appeal to the newly initiated cold war as a staunchly anti-communist regime. Following the Pact of Madrid in 1953, the Francoist regime was supported by American aid in return for bases and opposition to the USSR. This was much welcomed by Franco despite his nationalistic ideology, whose strict self sufficiency had not had the desired economic effects during the period of isolation following the war, instead stagnating. The Americans in return appreciated having a reliable ally in a very strategic location in Europe

Sources:

Preston, Paul. Franco and Hitler: the myths of Hendaye, 1940

Beevor, Antony. The Battle for Spain

Seidman, Michael. Workers Against Work

Thomàs, Joan Maria. Roosevelt and Franco during the Second World War

http://trove.nla.gov.au/ndp/del/article/48924187 Pact of Madrid