Server Speed = Port Speed?

Trying to figure something out here:

If I have a server that has 1 Gig/s Ethernet capabilities… and, for the sake of this argument, say it’s on a 1 Gig/s internet connection… would this server then be able to serve 1 Gig/s to the internet???

Is there a port switch or router that would limit its bandwidth performance? If so, what is this hardware called? (I suspect I’m missing a piece or two of the puzzle.)

I’ve been racking my head on this for several days now and I’m going crazy. I would appreciate any advice on sorting these semantics out so I can optimize my server’s data transfer rate capabilities.

Thanks!