Nowadays when browsing web with a mobile device, often you get a choice to use mobile version of the website. When you say yes, you will get redirected to this mobile version. Sometimes you get even redirected automatically based on your browser's user agent string. Usually this mobile version is having either different origin (most of the time just different hostname), but sometimes URL is modified some other way to indicate that you use mobile version.
Often there is nothing wrong with this, but things get annoying when you use something like Firefox or Chrome sync service that allows you to synchronize the state of your browser across all your devices. Now think what happens when you use your mobile device and want to continue with your desktop machine. Everything gets synchronized nicely, but then you get the annoying part. When you started browsing with mobile device you will get the mobile version even when later switching to desktop. Hardly ever this mobile version is optimized for desktop use. To make things even worse, it is quite common that the mobile version has completely different structure that does not map against non-mobile version. In such case, switching between mobile and desktop versions might not even be straight forward.
For the past few years now, there has been a trend in web design called responsive design. Idea is that you design your web site in a way that you modify the layout and behaviour dynamically depending on certain rules. You do not build separate version of the web site for different types of clients. To simplify things a bit, idea is to have have single index.html that will work for all kinds of clients and dynamically adjusts the layout to match the most optimal layout profile for the client. When building your web site or app properly, you can easily support more than just mobile and desktop layouts. You can have one for small screen phones, one for larger screen phones and small screen tables, third for large screen tablets, fourth for desktops and one more for large screens people watch from distance like TVs or billboards.
If you work for creating a web site or an app, go there and find out more about responsive design and make your product properly responsive. Make sure your web site's structure works well for different layouts. And keep in mind that regardless of the client type, URL should always stay the same, indicating the page or part correctly regardless which layout is used. This way you make sure that users get the best possible experience and can switch between devices.
To test your site's/app's design, you can use something like Firefox's "Responsive Design View". As an example of some basics of responsive design, you can check my notes app in GitHub. It is not very pretty, but shows how you can switch between handset, small tablet and desktop layouts mainly using plain CSS.