Local push
Delivering notifications over the local connection instead of the push service.
When your phone is on the home network, the app can hold an open connection to your server and receive notifications straight down it. Nothing leaves the house, and there is no third party in the path.
When it is used
Local push only applies on the networks you have marked as home. Away from home the app falls back to the push service automatically — you do not have to choose.
What it costs
Holding the connection open uses a little more battery than waiting for a push. On most phones the difference is small; if you are chasing battery life you can turn local push off in Settings → Notifications and rely on the push service.