If you have an Owner/Administrator Permission on the Site that you want to create a Team Site beneath, you can follow the instructions below. However, if you want to create a team site under a parent site that you do not have permission to do, you need to contact the site owners of that parent site to do that for you.
1) Go to the parent site that you want to create create a new Team Site beneath. Click on "Site Content" button
2) It will direct you to a page that shows you the list of content on your parent site. Scroll down on that page to get to "new subsite" button. Click on it to create a new subsite.
3) Put information of the new team site: Title, Description, Website Address. Select Team Site as a template. Set permissions, and Navigation Inheritance as needed for the new team site.
For Permissions settings: You can select unique permission, if you want the new team site to have different user(s) to manage it.
Tips: Browse for user button makes it easy for you to find the user you are looking for. You can search by using his/her GT Account, Email Address, Name, and so on.
4) Click on "Create" button