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We'll get straight to the point:
You can play Arma 3 for FREE on Steam until this Sunday, January 19th (10 AM PST). You simply need to visit the Arma 3 store page on Steam, and fire your mouse left-click at the green "Install" button.
To continue playing Arma 3 after the Free Week has ended, you can take advantage of a massive 70% discount on the Arma 3 base game, and/or enjoy savings of up to 66% on Arma 3 DLC packs and expansions until January 20th. When purchasing the Arma 3 Ultimate Edition (complete-your-collection bundle), you will seize an additional 15% bundle discount on top.
During the Free Week, all Steam users will have free access to the Arma 3 base game plus the Arma 3 Apex expansion. This also includes Arma 3's full "The East Wind" singleplayer campaign and the "Apex Protocol" co-op campaign. If you're new to the game, you might also want familiarize yourself with the game's controls via the "Prologue" tutorial campaign.
Here's the full list of discounts (January 14-20):