Innsbruck, in the middle of the Tyrol at 575 meters of altitude, has a continental climate featuring cold winters and sunny, warm summers. Innsbruck is famous for skiing, which is said to be best between Christmas and New Years. Check the snow conditions before you go.
Innsbruck, along with Salzburg, is known for rains called "Schnürlregen" ("rain in strings"), a very heavy rain that seems to fall in curtains of "strings" but is usually quickly over.
Innsbruck has quite distinct seasons; you can enjoy them all. Summer weather starts early in Innsbruck, and an early summer visit will probably turn out better than one in late summer, when you'll likely encounter more rain, as you can see from the rainfall chart below. Winters are cold and there can be lots of snow--but it's still a popular time to go to Innsbruck and there's usually plenty of sun with the chill. The Christmas market and concerts are a huge draw along with winter skiing nearby.
The charts below show the historic high and low temperatures and the month by month precipitation averages for Innsbruck, Austria.