First of all, to get your Ymail on your Android phone you must be able to get your Ymail via POP. The only way to do this is either to pay for Ymail Plus, or agree to receive a newsletter once a month (which actually never comes) to get Ymail POP for free. If you aren't doing either of these, you can't get your Ymail through the email app, you will have to use the mobile browser instead.
As for getting it to auto update, you must do the following:
In the Email app > Setting for your Ymail account > Setting > Send & Receive > you must set a Check Frequency. The shorter the frequency, the more battery usage.
The email app must be left running in the background. Whether you "back out" or press Home, it should still be running. If you want to check, get a task app like TasKiller to see what apps are running in the BG.
Your phone settings must be set to update in the Background.
Go to > Settings > Data Synchonization > Data Synchonization > and make sure that Background Data is ON. I think you might also need Auto-Sync on, but I'm not sure.
With these settings on you email app should be checking for email ini the background at the frequency you set. Obviously you must also have either WiFi access on (and connected) or access via your data plan.
BTW: Attention Mods! Just because this guy mentions he is using a Droid, doesn't mean it's a problem specific to Droid, and should therefore maybe be in one of the general topic areas, instead of the Droid area?