What is burst current?
ESC's have two specifications related to current, the continuous current and the burst or peak current. The continuous current, as the name suggests, is the current that you can safely draw for a long / continuous amount of time.
On the other hand the burst current is the maximum current the ESC will provide for a short amount of time following a cool down period. Unfortunately this time varies from brand to brand and might not even be consistent from one model to another - some vendors specify it with five, some with ten and some even with 25 seconds.
Pushing an ESC above its burst current rating can cause de-syncs and even damage it permanently, causing the quad to fall out of the sky.
The current you will draw from your ESC depends on the S rating of your battery, the motor and propeller combination. Usually you will find a current specification in the motors detail section.
To be on the save side do not expect the ESC to provide its burst or peak current for a longer time period than five seconds. Personally I like to choose my ESC's in such a way that I do not push them to their limits and rather have some buffer to the continuous current rating. This also allows me to be more flexible regarding motors and props, because even a change of props can cause the current draw to increase.