How to minimize battery usage while hatching eggs in Pokemon Go?

The easiest settings to reduce the energy consumption are your Location settings:

->> Settings ->> Location

->> Mode

Here you can choose battery saving which has quite a big influence. Your battery is saved at performance's charge: When I activated the "battery saving" location setting I noticed some huge latencies when walking around whereas I never faced those issues when I played with the "High Accuracy" option.

I think this option is available since Android 5.0. Correct me if I am wrong.

Another option is in the Application's setting itself:

->> Pokemon GO ->> "Pokeball" ->> Settings

I do not know what this option does exactly but I think it also tries to reduce the accesses of Android's Location service (Also look at this topic: What does the battery saver do in Pokémon Go?).

Update June 30, 2016: This Update removed the In-App Battery Saver for iOS.

Tip:

Always carry along a charger and hang around nearby sockets :P

Best is maybe also download the map in google maps for your area? I did this and it seems to improve the battery loss

In the battery saving mode, when you hold your phone so that the top faces the ground, the screen goes black with a very dim white pokemon go logo in the middle of the screen so it saves energy. When a wild pokemon jumps out, the screen will notify you and light back up.