As already mentioned, there are different interpretations of the causalities here.

After the Winter War, the Soviet threat was obviously a major factor in Finnish foreign policy. (Before the war, neutrality and the League of Nations had been assumed to be enough to prevent aggression.) As such, Finnish politicians tried to find help from wherever they could. The first choice was the Western allies, whose offers for help had, in part, influenced Stalin to agree to a ceasefire and sign the Peace of Moscow (1940) to end the Winter War before Allied troops could come to Finland's aid.

However, the German conquest of Denmark and Norway in early 1940 put an end to these plans. Finland was now practically isolated from the Western allies, particularly since Sweden refused transit rights for them (and it would've been unlikely for Western allies to ship troops through German-dominated waters anyway). The conquest of Norway also put an end to discussions about Nordic defence union, but, on the other hand, made Sweden actually more receptive towards a Swedish-Finnish union. The two prerequisites the Swedes had for such an union were Finnish renunciation of any revanchist designs regarding the areas lost in the Peace of Moscow, and an approval from both Soviet Union and Germany. Finland agreed instantly, but the Soviet Union informed in no uncertain terms in late 1940 that such an union would be a breach of the terms of the Peace of Moscow - and that was the end of the union plans.

Before this, however, the Soviet Union had been making quite threatening moves against Finland. During summer 1940, it moved combat-ready divisions to Finnish border, very nearly precipitating a general mobilization in Finland. The Baltic countries were in practice annexed, and Soviet fighters shot down a Finnish airliner Kaleva over the Gulf of Finland. A "Finnish-Soviet Friendship Society" became very active and began to influence Finnish domestic politics. The assessment of both Finnish and German intelligence was that Soviet Union would be ready to finish Finland from August 1940.

To this end, Soviet foreign minister Molotov met with Hitler and demanded free hand to deal with Finland, as per the secret protocols to the Molotov-Ribbentrop agreement. However, the Germans refused and vacillated, as by this time Hitler had began to turn his gaze towards operation Barbarossa. Molotov did not get the permission he sought, and from late 1940, Finnish foreign policy began to pivot towards Germany as the only feasible source of military and economic aid. (Finland needed food supplies in addition to weapons, and without German aid there would have been mass starvation in 1942.) This German turn was not without its critics, but there were few enough options left.

High-ranking Finnish officers were shown initial Barbarossa plans from fairly early on, and very likely took part to detailed planning as well. By about March 1941, Finnish army was more or less committed to Barbarossa, and although the Soviets realized their error then and reversed its stance about Swedish-Finnish union, by that time it was too late.

My interpretation of the events was that the Soviet threat after Winter War was extremely real, and it caused Finnish politicians to seek any support they could find. After Norway had fallen and the Swedish pact torpedoed, there was only one option remaining - and when that option was taken, the hopes for regaining Karelia and the second-largest city in Finland with German help trumped caution.

As for the strength of Finnish allegiance - it wasn't very strong. However, German forces held the entire Northern Finland, and conducted their own operations - with very little to show for their efforts - until 1944.

Finland had her own reasons for the fight, and these were limited to regaining the lands lost during the Winter War with some extra as a compensation, and to fulfil the nationalists' dream of uniting East Karelians with Finland. Despite repeated German entreaties, Finnish forces refused to cut the Murmansk railway, which was very important for initial Lend-Lease shipments, and flatly denied requests for help in the siege of Leningrad, although cutting the remaining routes to that besieged city would have been quite possible in 1941 or early 1942. Meanwhile, Finnish diplomats did their utmost to communicate the Finnish position to outside world, and had some success, as shown by the differential treatment agreed by the Big Three: while the Allied Powers required unconditional surrender from all the other German allies, Finland was to be the exception.

By early 1943, Finnish high command and political leadership were convinced that Germany was going to lose the war eventually, and this froze the cooperation even more. Finland made first peace feelers in early 1944, but got terms that would've amounted to unconditional surrender and occupation. Nevertheless, Finnish high command believed the Soviets wouldn't waste forces in attacking Finland - a major miscalculation that very nearly resulted to defeat during the summer 1944 Soviet offensive. This offensive, however, was ultimately repulsed with help from German troops and anti-tank weapons. As a price for this help, the German foreign office extracted a promise from Finnish president Risto Ryti: he promised that neither he nor any government appointed by him would sign a separate peace with the Soviets. After the Soviet assault was stopped, Ryti promptly resigned, leaving the next president - Mannerheim - to sign an armistice.

The stipulations of the armistice included a demand to force German troops from Finland. This led eventually to the "Lapland war" against the retreating Germans, and even though it began as a "phony war" of sorts and was just a skirmish by the standards of the Second World War, it was nevertheless a real conflict that saw over 900 killed in action and close to half of all buildings in Lapland destroyed.

The above is based on a fairly broad reading of various sources about this topic, but if you want one English-language work examining the Finnish relationship with Germany during the Second World War, it would definitely be Finland's War of Choice by Henrik Lunde.

See also my sort-of review in this thread.

There are really two schools of thought about this.

The first one is what was taught in Finnish history up until recently. Basically: the Finns were under threat from an invasion from the Soviets leading up to World War II. The border of Finland was something like 16 miles from Leningrad and Stalin wanted a buffer zone between the Finns and the Soviets. He also wanted control of Finnish ports and demanded control of them. The Fins did not oblige, and therefore the Soviets attacked, leading to the Winter War of 1939-1940. The threat of Soviet attack prior to the Winter War pushed Finland into an alliance with the Nazi's, who had already demonstrated that they wanted an influence in Scandinavia. In fact, unless I am very much mistaken, the Nazi-Soviet Pact originally implied heavily that the Soviets would be taking Finland.

In short, Hitler most likely found it advantageous to befriend the Finns as he intended to invade the USSR. The Finns were pushed into this alliance from fear of Soviet invasion, which eventually happened.

There is however debate over whether or not the Finns were willing contributors with the Nazi's and whether they knew about "Barbarossa" (the invasion of the Soviet Union by Germany). There is speculation that the Finns helped the Nazi's actually plan the attack, perhaps with knowledge gained from the Winter War. It was the failure of the Soviet's to take Finland in the winter that helped play a part in Germany's confidence heading into their invasion of the Soviet Union. Under this theory, the Finns are painted as being ideologically sympathetic to the Nazi's, the fear of Soviet invasion taking a back seat.