First off, close Terraria. You cannot directly download the Calamity mod via Terraria. Instead, search for tModLoader on Steam and install that.
Now that you've got the right software, launch it. You'll see that you have a few new options on the title screen. Select Mod Browser (not Mods). This will bring up a list of available mods. Up top, right next to where it says Mod Browser, there's a search bar. Search for Calamity in that bar. The list will update in real-time as you type, so no annoying loading times.
The mod you're looking for is titled Calamity Mod (No Calamity Music), and if you hover over the name, you'll see By: Fabsol and the Dev Team. Click the green download arrow and the mod will immediately start downloading. If you want, you can also download Calamity Mod Music made by the same group. The music is honestly fantastic and totally worth the few extra seconds of downloading.
Related: Best Music Packs in Terraria
Once you've downloaded the Calamity mod (and possibly the music as well), then click Back at the bottom of the screen. Once you do, tModLoader will do a slight reload, which will activate the Calamity mod. It will do this every time you start the game or download/enable/disable a mod.
Once it's fully reloaded, all you have to do to start playing is... well, start playing! Make a character, generate a new world, and have fun!
If at any point you wish to disable the Calamity mod, all you have to do is go into Mods from the title screen and click on Enabled to disable it. But be careful about loading up any characters or worlds that contain Calamity content while the mod is disabled.
For more Terraria mods worth considering, check out Best Terraria Mods!