The mode switch allows you to specify which locating methods apps may access to determine your location. This has implications for both battery usage and accuracy of location.
The high accuracy mode simply enables all methods for determining your location, including GPS, Wi-fi and mobile network location.
Notably, GPS is the most accurate method for determining location, but it is also slow and consumes a lot of battery power. GPS relies on receiving weak, precisely timed messages from several satellites orbiting earth. Using a combination of precise timing calculations and knowledge about where each satellite is, a reasonably accurate estimate of your location on earth can be formed.
The satellite signals do not penetrate buildings or other structures well, so accuracy may be poorer or the location may be completely unavailable when inside buildings, underground or in vehicles, however, being in the top story of an ordinary home or in a regular car should still result in a fair signal.
Note that there are more than one kind of positioning satellite orbiting earth, including GPS, GLONASS and Beidou. Your device may be able to use one, two, or all three of these, even if this menu only mentions "GPS". The more it is able to use, the greater the chance of finding an accurate position.
The power saving option enables all locating methods except for the satellite positioning (GPS) method.
The two other methods for determining position: wi-fi access points and mobile network location, both use considerably less battery power, but at the expense of (usually) providing lesser location accuracy. In cases when you are in a remote location with no nearby wi-fi access points or mobile coverage, you won't be able to determine your location at all.
This option enables only GPS, but not the other methods of determining location. In "high accuracy" mode your device will always prefer a GPS signal anyway, and only use other methods when GPS is not available. This option disables that fall-back mechanism.
Another thing worth noting is that some apps only request a low-accuracy position, and when this option is set those apps will not be able to determine their location at all.
The method of determining location via wi-fi access points (which is enabled in both "High accuracy" and "Power saving" mode, but not "GPS only" mode, necessitates periodically sending information to Google's servers in order to look up the location from their database of locations of wi-fi access points, or to allow Google to improve their database. This communication may happen even when no app is requesting your location. While your device does communicate some data that could be used to determine your location to Google, Google does not intend to store this data or associate it with you personally. However, you may feel that you do not want this information going to Google at all, in which case you may select the "GPS only" option.