Mail opens up each time I turn on my computer. How do I stop this?

I have been using my mail icon to open my mail account through my local cable provider.

I was told today I should no longer do this but to access my mail account through the web. mail.bex.net

This worked out fine.

Problem. I cannot send out an e-mail. The system will not access my mail directory to send out an e-mail.

Each time I turn on my computer my mail account opens automatically from the mail icon under the for system.

Hcw can I turn off my mail icon turning on each time? I think this will allow me to turn on the mail directory when I use the web.

Thank you.