If you're a business, you probably find it annoying and would like to add some rich text linking to your website or display your logo.
There are instructions online on how to do this, but eventually, ios seems to break the images (since they aren't really supported).
Now there's a solution that works, even if you've closed the Mail app or even turned your device off. Although this does require some knowledge of html and saving files - you probably already know how to accomplish that, since you're looking to add an html signature in the first place.
Create your html signature on your desktop computer, either in an email program, using the html tools available in your regular email or webmail client or using an html editor.
Important Tip: Don't paste images into your signature, upload them to a server and link to them. This prevents being blocked by email programs for 'dangerous' embedded content.
Save the signature as an html file by copying the code into a text editor like notepad and save the file as something like 'signature.html'.
Upload the file to your favorite cloud server (like dropbox), or email the file to your ios device as an attachment.
On your ios device, open up your dropbox or other cloud program and view the file, or open up the html file you attached and mailed to yourself and view the file.
When viewing your signature html file, select and copy it.
Open your ios device settings, tap on 'Mail, Contacts, Calendars', then scroll down and tap on 'Signature'.
Choose the account, or select 'All Accounts', delete what is currently in the text box and then paste the copied html signature into the box.
That's it. Try it out and see how it looks and enjoy!