These links are tied to the hardware – the computer you are working on. If you add a site as a favourite in a training centre using their computers for example, you won’t have those links when you go back to your own computer.
2. My Links
Links added here are tied to your user ID. That means that if you go to another building or region and log in to SharePoint at someone else’s workstation, you will still have all your links available regardless of what site you are on. They may or may not be visible to other people depending on how you set them up. Links added here will also be syndicated to your My Site profile page.
3. Team Links
These links are tied to the team site and are visible to everyone who has access to that site. They are links you wish to share with everyone in the team. You can have multiple sets of links, one for research maybe, tutorials, industry standards, etc. These links will unfortunately not open in a new window, therefore teach people to right-click and open in a new window so they don’t lose where they came from. If you need your links to open in a new tab by default, delete the Links web part and add the Content Editor Web Part instead. Links added there have a setting option to open in a new window.
Make sure you add your links in the correct place or you may not have access to them.