The 1 Gbps means Port Speed you will be receiving for the server. This shows your available maximum download/upload speed between the server and you. It is most important when you have a lot of files to upload and download, then having 1 Gbps increases the speed for you.
It is also important to note that this not necessarily means that you will be getting this much speed, as it depends on your internet service providers and network limits. Also, before ordering this kind of server with 1 Gbps I recommend to ask the provider if it is shared or just for you.
In addition to the previously mentioned considerations, it is important to verify whether your 1Gbps port connection offers full-duplex connectivity. If so, then you should expect to be able to upload and download at 1Gbps simultaneously, important for busy servers.
@Kaz Wolfe Do you know what does 1 Gbps (1000 Mbps) physical interface mean? it is standard for most servers and to most switches. The servers are usually connected to 1 Gbps ports, but what kind of measure data transfer (in TB) or what kind of bandwidth (in Mbps) would one use depends on the contract one has with the web host. There might or might not be logical limits on the 1 Gbps interface.