What's better than a banner? Why, an animated banner of course. Animated banners are a great way of catching extra attention, and make you website more dynamic. They are mainly useful for home page, as once your user is engaged, you want them to engage with your content. Animated banners are 2D banners, meaning you have your usual banner background, but it's written content and elements are animated in some way. And the important part is, written content is not a part of an image, rather a proper text which can be indexed and adds to your content.
Banners have many uses, and their uses are generally dictated by the page the banner is on. They're designed to attract attention, which isn't found in abundance with users. People mainly look for something specific, and don't necessarily care or pay attention to banners as much as they used to. And with all the endless advertising we're all bombarded with on daily basis, we're slowly gaining immunity to it.
On the homepage, the most important banner is usually found. And this is the one that can benefit the most from it being animated. Make it count, it should be interesting, and relay something the user can benefit from. If this banner is not hitting the mark, chances are that others on your site won't even matter.
With shops, only other pages an animated banner would be of use on, are category pages. This can be to alert the users to a special you're running, or that you have free delivery on products in this category, or whatever the case may be.
For shops product pages should be clean and devoid of any distractions from the product they're looking at, so I wouldn't recommend any kind of banners on those pages.
For info sites however, it's different. Since such sites tend to be information heavy, something to break up the design, the content and the headings, can be beneficial and enjoyable. But they should remain static banners, and ideally reflect some meaning of the page they're on, or a general one that's on all pages, reflecting the site itself.