When I open apple mail on my ipad 3 icons appear at the bottom of the screen. I don't want them to automatically appearon my screen

You have now encountered the iPadOS “Shelf” - this being a feature of the revised iPadOS multitasking environment. 

iPadOS permits you to have multiple “instances” or windows of the same App - each instance/window being self contained. The shelf, when triggered, shows a thumbnail of each running instance/window of an App. 

For example, if you have three running instances of Safari, each with its own set of open tabs, you will see three thumbnails corresponding to each running instance of Safari - plus a button to open a new instance of Safari. Simply tapping a thumbnail will switch directly to that running instance of Safari. For Mail, you may see a separate window for open messages, drafts, or mailboxes.

Each App - such as Mail, Safari, Notes, Calendar, etc. - has its own corresponding shelf.

Individual open instances of an App can be closed directly from the Shelf; just swipe the unwanted thumbnail off the top of the screen. Note however, that Multitasking and associated Shelf functions cannot be disabled.

This article from Appleinsider has a useful explanation of this feature along with many of the recent updates to multitasking:

https://appleinsider.com/articles/21/08/12/how-to-master-multitasking-on-ipad-and-ipad-pro-in-ipados-15

The iPad User Guide is also a rich source of frequently overlooked information. The Guide is available both online using a web browser (such as Safari), or as an Apple Books download:

iPad User Guide - Apple Support

https://books.apple.com/book/id1567104892

Unlike iPadOS 14 and earlier, multitasking cannot be disabled since introduction of iPadOS 15.

The iPadOS Shelf was introduced with iPadOS15 - and is an integral element of iPadOS multitasking.