Finland was the only democracy that fought on the side of the Axis during the Second World War. It's extremely certain that without the Winter War, Finland would've remained neutral like Sweden, and except for a small group of Finnish Nazis, Finns in general did not see an alliance with Nazi Germany as a desirable state of affairs. Nazi ideology was viewed with distrust or outright revulsion by large segments of the society, key political decision-makers included: in fact, the Social Democratic Party ministers probably understood that if Germany won the war, Finland would most likely become a puppet state and Social Democrats would be hounded to extinction. Before the war, Finland was generally pro-British, and the Finns identified as the "Outpost of the West" and with the "Western Democracies" such as Britain, France, and the United States, not with dictatorial states in Central and Eastern Europe. A famous and oft-quoted 1931 poem "On the Border" by Uuno Kaila went something like this (my own translation):

Border gapes like a chasm.

Ahead, Asia, the East.

Behind, Europe and the West;

Guarding it, as a sentinel.

Finland's decision to join forces with Germany in 1941 was to a significant degree a result of all the other options failing, though with just enough political incompetence, desire for vengeance and greed thrown in to not totally absolve Finnish politicians for their decisions. After the Winter War, Finland's position was extremely precarious and there was a common, and very well founded, fear that the peace wouldn't last and Stalin would finish Finland at his convenience. This is precisely what Stalin intended, and during summer 1940 the Soviet foreign minister Molotov specifically asked Hitler in a conference in Berlin for "free hand" over Finland. The Red Army massed along the Finnish border, very nearly precipitating a general mobilisation in Finland, and incidents like Soviet fighters shooting down a Finnish airliner Kaleva over international waters raised tensions. The problem for Finland was that the Moscow Peace of 1940 had robbed Finland of the most important defensive lines along the Karelian Isthmus, and the army was still making up its losses after the Winter War. An attack during summer or autumn 1940 would most likely have breached the Finnish defences relatively quickly, particularly given that the Soviets had learned from the Winter War's early mistakes.

However, even though the Molotov-Ribbentrop pact was still on, and even though Hitler had given Stalin a free hand earlier (even to the extent of blocking arms shipments to Finland from crossing German territory), by this time Hitler's plans towards the Soviet Union began to solidify. So Hitler refused.

Nevertheless, the preferred course of action for Finns was a defensive pact with Sweden. Finns were so desperate for the pact that they agreed on all the terms the Swedes demanded, including a public disawoval of any "revanchist motions" or other attempts at regaining territories lost during the Winter War. However, the Swedes also required the acquiescence of both Germany and the Soviet Union. Both refused, Germany because Hitler saw that lone Finland would be an useful ally in the coming fight, and Soviet Union because Stalin still harbored designs on Finland. So by 1940's end, Finland was left with one option only: military help from Germany, no matter what strings came attached.

The Soviets realised their error in late spring of 1941 and signaled they'd be willing to let the Swedish-Finnish pact happen, but by that time, the Finnish politicians and military high command had quietly committed themselves to a rematch with the help of hitherto victorious German arms (recall that the loss of Karelia was an extremely deep wound and had resulted to nearly 400 000 internal refugees). German troops were quietly moved to Lapland in preparation for an attack towards Murmansk, and Finnish military began to receive armaments from Germany. The German troop concentrations in Lapland were kept secret (the official cover was that Finland just allowed the passage of German troops going to R&R from Northern Norway), and outside military high command, very few people were aware that Finland had de facto committed itself to Operation Barbarossa.

However, because just about everyone distrusted the Nazis - after all, they had taken over Denmark and Norway - and because of strong pro-British and pro-Allied sentiment, Finland did not want any formal alliance. All the Finnish politicians and Marshal Mannerheim very clearly stated that Finland would not join the Tripartite Pact (the original Axis pact) and would fight only to the extent of redressing the territorial wrongs caused by the Winter War.

When the Germans struck on 22 July June, the Finnish military had been mobilised but was occupying defensive positions along the border. However, German and Finnish minelayers began laying mines on Finnish and international waters, and Finnish airfields in Southern Finland recovered and refuelled German planes returning from raids over the Soviet Union. In response, the Soviet Air Force bombed several Finnish airfields and other targets, and on 25 July June, the president prime minister of Finland declared in a radio speech there was no other alternative but to fight "a defensive war." The Finnish foreign legations were instructed to emphasise that Finland would fight a separate war to regain territories taken by force, and it just so happened that Germany had the same enemy. This was undermined to a considerable extent by Hitler's declaration that Soviet Union was being attacked by Germany and her allies, from Finland to Bulgaria; however, the term preferred by the Finns was "brothers in arms" instead of allies.

The extent to which this distinction was appreciated by Allied nations differed: pressured by Stalin, and after not receiving sufficient guarantees that Finland would withdraw from the war, Churchill had the United Kingdom declare war on Finland on 6 December 1941. Most Allied maps from the period that I've seen depict Finland as an Axis ally and don't make any meaningful distinction, and I believe few people saw much difference. However, the United States never declared war against Finland, and even maintained diplomatic relations until 1944. The official US policy was that Finland had been forced to fight the war. Perhaps most importantly, Finland's special status was recognised in the Casablanca and Yalta conferences, and Finland alone from all the Axis allies was permitted to choose her own government post-war. After the Soviet assault of 1944 had been stopped without Finland being overrun, this also happened.

For the most Finns, the preferred outcome of the war would've been Germany destroying the Soviet Union but exhausting itself so that the Western Allies then destroyed Germany. This did not happen, but there was genuine relief that the Allies and not Germany emerged victorious, even though it was mixed with the pain of defeat and permanent loss of Karelia. Here's a picture of Helsinki's central Railway Square on 10 May 1945, as the news of Germany's surrender came through: the people are celebrating.

https://pbs.twimg.com/media/DJtFn7qW4AE-HPU.jpg:large

(Source: Helsinki City Museum. Photographer Väinö Kannisto.)

EDIT: some corrections.