It depends on what you consider the "Axis". There are generally two treaties that can be seen as instrumental to the formation of the Axis. One is the anti-Comintern pact signed in 1936 that was targeted against the Soviet Union and Communism in general, the original agreement was between Germany and Japan and Italy signed a year later. This was the original "Axis". When the pact was renewed five years later in 1941 it included more nations including Finland. But a bunch of other neutral nations like Spain and Turkey signed the anti-Comintern pact. Finland did not however sign the tripartite pact, which pretty much every Axis nation signed.
Immediately after the Winter War in 1939 (in which Finland lost territory to the Soviet Union) the Fins began to cooperate with Germany militarily and granted military bases to Germany. This is generally seen as the start of full scale cooperation between Finland and Germany. When Germany launched Operation Barbarossa, Finland launched what is often called the "Continuation War" Finland wanted to get its territory back but also wanted to walk a "neutral" path. They didn't want to cooperate with Germany unless they absolutely had to, and above all they wanted to make sure to stay independent and not become another German puppet state.
To this end the Finnish commander in Chief Carl Gustav Mannerheim, only launched a few offensives, most of which were simply to recover lost territory, there was one major offensive they did launch with Germany against the Soviet port of Murmansk, but it didn't work out. Finland refused to help the Germans capture Leningrad (which frustrated Hitler massively). Finland did manage to protect its Jewish population and did not fall under the sway of Germany the same way some other Axis countries had, though many Finns ended up serving in the SS.
Anyways, the Finns basically sat tight until the Red Army pushed them back in the Autumn of 1944, Mannerheim saw that his country couldn't continue and sued for peace. The Finns ended up losing more territory and they had to force the remaining German troops out of Finland.
As for the original question, it depends on who you ask. Finns will vehemently deny that they were part of the Axis and claim that they were "Co-Belligerents". But really the truth is they were part of the Axis. A very independent member, but a member none the less.
You can find what being a memeber of the Tripartite Pact entailed here:
http://avalon.law.yale.edu/wwii/triparti.asp
And here is what being a member of the anti-Comintern pact entailed (the one Finland signed)
This is going to depend on how pedantic you want the answer.
The Axis powers were those who signed the Tripartite Pact. This was originally between Germany, Italy and Japan. http://avalon.law.yale.edu/wwii/triparti.asp
Other countries would end up also signing the Pact, such as Bulgaria and Romania. However, Finland was not one of them. It did however fight on the side of Germany against the USSR due to the Soviet Invasion of Finland. http://www.ushmm.org/wlc/en/article.php?ModuleId=10005177
So, they were not officially a signatory of the Axis. However, they did fight alongside the Axis during the Second World War and are counted as a co-belligerent.
hi! you'll find some more info in these previous posts
Was Finland an Axis state during WWII?
Why is Finland not more closely associated with the WWII Axis powers in history texts?
What was up with Finland during WW2?
To what extent did Germany and Finland collaborate during The Winter War/World War II?
I haven't included any posts on the Winter War (between the Soviet Union and Finland); if you're interested in that also, just run a quick search in this sub for "Finland war" or "Finnish war"
Another dimension to the previous informative response by panzerkampfwagon goes back to WW1. Prior to the Russian Revolution, the Grand Duchy of Finland formed a very significant part of the Russian empire. During the Russian Revolution, the Fins fought a civil war between sides that aligned roughly with the Bolsheviks and the German Empire. The 'conservatives' won and the Germans provided them with considerable military support.
Mannerheim, who was both a staunch defender of Finnish independence, and a military strongman with significant popular support, had to try to keep Finland independent of both Soviet Russia, and Hitler's Germany. Although Finland lost some big chunks of territory, and although she was a de facto German ally, emerged from WW2 still independent.
[This is no more than a linkless wiki of a description. The narrative is very general and offered only as additional background.]