Well, text messages should always be pushed.
For email, there is two ways to get email on your phone, push or pull. If you use the Gmail app with a gmail account, it is push. That means when an email gets to your gmail server, the server pushes the email to your phone immediately.
Other email types, like a yahoo account, must be pulled. That means the phone logs into your email account at a predetermined time (can be every 5 minutes to once a day) and checks for new emails. Obviously, pulling emails uses WAY more battery, because it is constantly using the internet to check for new emails, where push email uses literally no battery.