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.
If you have a busy download site (with many concurrent downloads) or a bandwidth intensive site (such as a video streaming site), having a server with 1 gbps can help you with fast loading and access time.
The larger the better. Though, you might want to ask your hosting provider to see if there's a limit on it as others have said.
The higher the speed, the better for bandwidth intensive site with huge traffic. If you don't have a site that requires a lot of resources, you might want to scale down on resources used till you have a need for something bigger to save you some costs.
I would add that in case you expect to have a stable traffic every day then it's better to search for the pricing per Mbits (because your traffic structure is the same all the time) and you do not overpay. If your traffic vary during a month (typical for short streaming events), I recommend to choose billing per transferred data (TB). Just ensure you have enough bandwidth capacity available to your server and then again you do not need to worry to overpay anything extra or that you will be saturated in peak time. 1Gbps dedicated bandwidth capacity should be definitely enough for all starting projects. If you are not sure, ask about trial period, some providers do so.