Open Gmail app from my app

You need to use custom URL Scheme. For gmail application its:

googlegmail://

If you want to compose a message there you can add more parameters to this URL:

co?subject=Example&body=ExampleBody

You can determinate if any kind of application is installed using this code (just replace customURL obviously for an other apps):

NSString *customURL = @"googlegmail://";

if ([[UIApplication sharedApplication] 
canOpenURL:[NSURL URLWithString:customURL]])
{
  [[UIApplication sharedApplication] openURL:[NSURL URLWithString:customURL]];
}
else
{
  //not installed, show popup for a user or an error
}  

Setup for iOS9+

As explained here, if you're on iOS9+, don't forget to add googlegmail to LSApplicationQueriesSchemes on your info.plist

Code to open GMail

Then, you can do the same as the accepted answer (below is my swift 2.3 version):

let googleUrlString = "googlegmail:///co?subject=Hello&body=Hi"
if let googleUrl = NSURL(string: googleUrlString) {
    // show alert to choose app
    if UIApplication.sharedApplication().canOpenURL(googleUrl) {
        if #available(iOS 10.0, *) {
          UIApplication.sharedApplication().openURL(googleUrl, options: [:], completionHandler: nil)
        } else {
          UIApplication.sharedApplication().openURL(googleUrl)
        }
    }
}