Mail can arrive on your device in a few different ways...
Manual: only get new mail as and when you ask for it to happen, e.g. Dragging down your inbox to kick things off.
Fetch: every so often, in the background. You decide how often.
Push: mail gets pushed immediately onto your device by the sender. iCloud supports this, and it's probably what you want.
Until recently, Apple were banned from pushing iCloud Mail in Germany because of a legal spat. Now, the lucky Germans can have push if they want it.