If your phone is an iPhone you can simply search "Snapchat" in the App Store. The first one that pops up with a yellow background and a yellow ghost is the official Snapchat app. Once you find it you simply hit the "download" button.
The same thing pretty much goes for Androids. You simply go to the "Play Store" app on your android and search for "Snapchat". Again the official one will have the same background and be the first one that pops up. Same with iPhone's you hit the "download" button to download the app.