Prior to Android 4.4 (KitKat), apps that are too large for internal storage could automatically load themselves into external storage, as long as the app supports loading into external storage (not all do!). That doesn't mean you would never run out of internal storage, though. It's quite possible, between apps, caches, and stored data that you could not have enough space even to download the APK. You have to manage this yourself by moving apps to external storage.
KitKat changes a significant portion of this by making the SD card essentially read-only, assuming your device has enough internal storage to do what you need. Most devices come with some sort of file manager that will allow you to move photos, videos, and other files off to the SD card, but most apps cannot do that themselves. Instead, a directory within internal storage is used by default. More importantly, as user GiantTree notes in a comment, even apps no longer get moved to the external card.
Adding an SD card doesn't magically expand memory. It's a separate storage area that you usually have to manage, especially with KitKat and probably Lollipop.