Re install Terraria on new laptop

Download Steam from this link - http://store.steampowered.com/about/

Run the file you downloaded. Log in using an existing account and enter your username and password for Steam. You may need to go to your email and enter a security code due to you logging onto Steam on a new computer.

After you log onto Steam, go to your Library and click the dark entry on the left labeled Terraria. Once you are on that page, click the blue Install button on the right panel. Keep clicking next. Terraria will automagically and install.

If you want to remove Terraria on your old laptop, go to Steam on that computer. Right click on the entry Terraria in your Library, and select the option Delete Local Content. Keep clicking next. After you click finish, it will delete all Terraria's files from your computer.