Connect Ethernet Switch to Gateway

I only have one ethernet connection at this location.

I don't understand that.  You say you have a BGW210-700 (fiber ONT wall connection to the red ONT socket on that gateway).  It has 4 ethernet jacks (yellow.  One of them should be tied up for your dvr (if you have a tv package).  If you have any wireless receivers than another is tied up connected to the WAP.  That leaves 2.  Are you already using one of those (a computer for example)?  If so you can use a gigabit ethernet switch on either or both of those.  It's best to keep the dvr and WAP directly connected to the gateway.

The switch plugs into one of the yellow ethernet jacks and you connect your ethernet devices to the switch.  Nothing needs changing in the gateway's configuration.   I suppose you could reboot the gateway after hooking everything up just to ensure its devices tables get updated but I don't even think that is needed.

To clarify, I am out of ports on the gateway and I need additional Ethernet connections in my home office.  My gateway is located in my wiring closet on the opposite end of my house and I really don't want to run more cat5 from this location to my office.  The additional connections are needed for hardwired devices.

Also, why do you say it's best to directly connect the DVR and WAP to the gateway?

Multicast IPTV traffic on the same switch as other ethernet traffic has the potential to cause problems.  There's a recent thread about this.  The GS108E may work.  Check that thread.

To clarify, I am out of ports on the gateway and I need additional Ethernet connections in my home office.

I'm still not 100% exactly following what you're trying to do so I'll list 2 solutions below. I feel that it will be my #2, but i'll list the other one anyways.

1. If you're trying to get other jacks in the house hot with data and you're all out of room in the modem's switch, then use one of the modems ports with a 4 foot ethernet cable to a switch, then plug any and all room's ethernet into the new switch. Keep in mind what xyzzy was saying, you want to make sure Uverse TV boxes are plugged directly into the modem.

2. If you're trying to simply just get more ethernet ports in a specific room, for example an office. Have the ethernet come out of the wall and go directly into a switch, thus giving you more ports depending on what size switch you get. Again keep in mind, if you have a Uverse TV box in the office using a hardwired ethernet connection a swich could cause issues.

As long as the switch is rated for 1gbps, and your using 4 pair ethernet cables your speed won't be limited by 100mbps.