Spain entering the war would have led to a temporary exchange of territory between the UK and Spain, as Spain and Germany take Gibraltar whilst the allies take the Canary Islands. The loss of Gibraltar might have made the North African campaign and Mediterranean naval war more difficult, but not significantly so.
Then we get to the end of the war. There's a case for Spain being a weak underbelly into Europe compared to France, with a weak Spain becoming a drain on Axis resources and an eventual path into France for the allies. So Spain's entry could have possibly changed the way D-Day and subsequent events unfolded, but I suspect the terrain is too difficult and Iberia too far away for such an alternative European landing to make much sense. However a landing in Catalonia could have been a counterpart to the landing in Italy... the way into France is easier on that side of the peninsula, I think.
There's no telling what the consequences of entering the war would have been for the Franquist regime itself. Maybe it would survive the war, but given that they would have inevitably lost, there could have been a coup against Franco like what happened in Italy against Mussolini.
What happens when Germany loses? Spain entering the war would have gotten the regime irrevocably identified with fascism, essentially confirming the widely held view that the Civil War was a rehearsal of WW2, and the view propagated by the Republicans that the Franquists were pure fascism... and I can't see the allies allowing Franco's regime to remain in power for this reason alone. Even though the war may have conceivably ended with Germany defeated but Spain not being invaded, I'd assume there'd be a surrender and occupation.
Assuming Franco is actually defeated and removed, rather than ignored and left in power after an armistice, the rest of Western Europe becomes a pretty good guide as to what happens next. Basically, the post-war would probably have been similar to the situation in France or Italy - strong left-wing parties, fascist parties (the Falange) banned, some regime elements becoming rehabilitated as conservative patriotic or Christian democratic parrties. However, given the terrible state of the country, I suspect that politics in a post-war Marshall Plan-receiving Spain would have been less smoothly democratic than in other West European countries: maybe following something closer to Latin America's development, more authoritarianism, greater risk of social revolution, etcetera. The social cleavages that played such a role in the Civil War weren't really healed, the country was still very poor and unindustrialised, and dominated by extremely conservative and elitist aristocrats and churchmen.
There's every chance that independent Basque and/or Catalan states may have resulted in the ensuing mess, given that these parties and movements would have been reliable, relatively conservative, non-communist anti-fascists. As for the rest of the country, the moderate conservative analogue to the Christian Democrats in Germany, or the French MRP, would probably have to be people like CEDA - the Catholic centre-right umbrella party from before the war, who backed the Franquists in the civil war but before that were relatively democratically minded. Maybe they would re-emerge as something prefiguring the modern Partido Popular, as eventually happened after Franco died. Maybe former CEDA head José María Gil Robles returns from exile and becomes the Spanish Konrad Adenauer.
Monarchy or republic? Open question, could have gone either way as in Italy.
