To deploy your new sublayout correctly you should create a Sitecore Package. This is basically a zip file that allows you to move both items and disk files between Sitecore instances in a controlled manner. For basic installs of Sitecore, where you have not added any specialised tools, it is generally the preferred way to move resources between servers.
The "Package Designer Guide" on the Sitecore Developer Network will give you information about how to use the Sitecore UI on your development site to create a package containing both the Item(s) and the file(s) for your sublayout:
http://sdn.sitecore.net/upload/sitecore6/65/package_designer_admin_guide-a4.pdf
Once created, this package can then be imported onto whatever other servers you want to deploy your sublayout to.
-- Edited to add --
Derek Hunziker's answer makes a good point: As well as the basic Sitecore behaviour there are third party tools available which can enhance and extend the deployment experience if you wish. As well as Hedgehog TDS, you might also consider:
The "Sitecore Rocks" extension for Visual Studio allows the creation of packages from within the
Visual Studio UI. This tool is free to use. (https://visualstudiogallery.msdn.microsoft.com/44a26c88-83a7-46f6-903c-5c59bcd3d35b/)
There are also a variety of open source tools - Sitecore Courier is one example: (https://github.com/adoprog/Sitecore-Courier) This is designed to help automate deployment between Sitecore instances.
Both TDS and Courier are most suited to regular deployments, such as those during ongoing development cycles, since they both include automation to help decide what gets deployed. The standard Sitecore UI and the Sitecore Rocks extensions for package creation are better suited to ad-hoc deployments, since you generally pick the things to deploy manually.