There is no official way to find out how many times an app has been downloaded from the iTunes store, unless you are the developer of the app.
Unlike Google play store, apple iTunes store does not reveals the app downloads count publically to all.
However, if you would like to know it, there are few unofficial ways, which will provide you with an approximate number:
1. Xyologic.com – a free service which offers estimates for app installs of any app you type in the search box. It provides downloads count for iPhone, iPad, Android and Windows Phone 7. [via]
2. Here is a formula to get a rough estimate of sales numbers of paid apps. [via]
Number of Ratings * 30 = Sales Estimate
This unscientific formula comes from the developer of River of News, a Google Reader app for iPad, who confirms the formula is reasonably accurate.
