We recently switched from iPhone to Android and on both of our Samsung phones, apps that we have notifications for (Facebook, Skype, etc.) do not have a number next to the icon showing how many notifications they have. They used to on iPhone, and the system apps on Android still do like the Messaging and Email apps. I've triple checked the apps' settings and turned everything off and back on and still have no clue why it's doing this
Android doesn't have "badged" icons like that by default. Unlike iOS, Android has had a pull-down list of notifications since the start, so that has become the standard way to present this kind of information in this platforms. Most Android users would be disappointed to see a 3 on their email icon, because they've come to expect a notification in the notification bar, giving more information than that. This is even more true since Jelly Bean added the expandable notifications: you can see the subjects of all the new e-mails, or, if there's only one, see a preview of the whole e-mail.
As eldarerathis points out, some apps offer a widget that you can use instead of the usual icon on your home screen. The widget might have an icon badged with a number of messages, as on iOS, or the app might instead offer a scrollable list of messages, the avatar/photo of the sender, etc., which isn't possible on iOS.
As you've noticed, Samsung phones can have badged icons on the default home screen. This is a feature that Samsung has added to make their phone look more like an iPhone. But since it's only in their home screen, not a standard feature, only their built-in apps support it. This is in addition to Android's usual, information-rich notifications, not instead of, so you're not missing out by having a Samsung phone.
If you've been an iPhone user for a long time, you might have to get used to using the notification tray to see new messages instead of looking at the app shortcuts, or to adding widgets to your home screen if you like, but once you do you'll find you can get at your messages with fewer taps and less fuss.
Badge based notifications are generally handled by whatever application Launcher you have installed. Google Now stock launcher by default does not handle this. Nor do many other launchers. However, Samsung's launcher does handle most applications but not all.
Nova Launcher has an add-on called TeslaUnread which can be installed in addition to the Launcher which will handle all badges for all applications. It can also be completely customized to have the badge appear in any corner you wish as well as color of the badge. Alternatively you can install widgets, such as Notifyer which will create a widget that appears as whatever icon you have chosen for whichever application and places it onto your launcher desktop. Whenever a notification is received, Notifyer will place a badge onto that icon.
I use a Samsung Note Edge. Samsung has an entire SDK for developers to create badges for their apps that are compatible with the Samsung launcher. As I noted before, Samsung has badges but not for every app. This is because apps have to have been custom designed with Samsung's badges in mind for them to function within a Samsung launcher. Personally I use Nova Launcher since it allows a lot more customization and control. Also I do not like the red or orange badges and preferred blue and Nova gave me this ability to change everything to how I like.