In Office 365 the first SharePoint site a user sees is called the Team Site, accessible from a link at the top of the Microsoft Office 365 portal site. The site is located at https://YourDomain.SharePoint.com, where YourDomain is the domain name that you use for SharePoint Online in Office 365.
By default the Team Site does not display a list of sites the current user is authorized to access, which some users may be accustomed to from the previous version of Microsoft Online Services in the so called "My Company Portal".
To help your users with such a personalized list of sites they can access from your companys Team Site, you can add a web part to display a list of site collections which are security-trimmed based on the search results for the user visiting the site.
1. To start out open your Team Site and put the page in edit mode (maybe add some explanatory text to the page)
2. Click the "Insert" tab under the "Editing Tools" tab - and then click the "More Web Parts" command
3. In the "Create" dialog click "Search" under "Filter By:" in the left part of the dialog, click the "Search Core Results" web part and then the "Add" button to the right of the dialog
4. The web part is added to the page (where you placed the cursor)
5. Rest the cursor at the upper right part of the web part to show the web part menu and then click "Edit web part"
6. In the editing dialog displayed to the far right of the screen, navigate to the "Results Query Options" area
7. In the "Fixed Keyword Query" text box, type Contentclass:sts_site, and then click OK
8. After clicking "Save and Close" at the top left, the web part now lists the sites the current user is authorized to access
How does it work in SharePoint 2013? I can't find the "Search Core Results" web part in SharePoint 2013 at all.