It's a storage, but yes it's part of the cycle also.
A forrest stores carbon as long as the forrest exists, dead trees emit carbon as they die/rot/burn away, living trees store carbon as long as they are alive (and they help maintain the soil where carbon gets stored also)
Once a forrest reach maturity , it will be a store like any other carbon store (but perhaps a fragile one , it can easily be cut or burned down).
Basically, if you remove a forrest (e.g. burn it down), most of the carbon is released into the atmosphere - thus it's a storage keeping the carbon it stored away from the atmosphere.
Forrests store about 400 billion tons of carbon in total , and we emit about 9-10 billtion tons of carbon per year.
(most numbers you find will be of co2, but those 2 oxygen molecules account for quite a bit of the volumn/weight also, so make sure to not mix numbers that account for just carbon with numbers that represents co2 )