iOS Wifi Connections - Search Domains
A Search Domain is simply a convenience that allows the system to convert host names to Fully Qualified Domain Names (FQDN).  Anyplace you might use a FQDN, such as in the location bar of Safari or as an argument to ssh, you can use a simple host name.  If the system can resolve that host name to an IP address (e.g. localhost -> 127.0.0.1) then it will use that.  If that doesn't work, then the system will try appending the search domain to the host name and try again. If you specified more than one search domain, it will try them each in order. 

If you are typically connecting to other Mac's on your LAN, then setting the search domain to local will make it easier to use just the other computer's names everywhere.  If you are in a corporate environment and all your servers are available at name.example.com, then setting a search domain of example.com will allow you to access them by just using name.

Those are the two most typical use cases.  When using a public Wi-Fi network, you should leave the Search Domain blank/empty to avoid accidentally connecting to some computer you didn't mean to connect to.