Lane priority is a basic to medium level concept you need to understand as a jungler. It will allow you to play correctly around your lanes and thus win more games.
What does Lane priority means?
It means the laner with "Lane prio" can respond quicker if the junglers are fighting (over scuttler or jungle camps). Usually the laner that can get to the fight faster is the one who is pushing.
Which lane has prio in this minimap? Top, as you can see Nasus can barely get out of tower (and if he does he will lose farm) while Irelia can quickly rotate to a jungle fight. Elise has the options to take the enemy jungle camps or Rift Scuttler when it spawns.
How to play around Lane Priority?
The most important thing you will get of playing around Lane prio is that you will minimize risks. For example:
This is the minimap of the same game, Rammus is doing a full clear that will end in his top side. He knows Elise started in her Red buff, so she will be top side too. Should Rammus even attempt to get top side Scuttler? No, enemy top has lane prio while mid is equally matched. Not only Rammus should not even think about Scuttler, but he also should be really careful while farming his top side camps, since Elise is an early game jungler and Nasus could not really help.
In the other hand, Elise, knowing she has both lane prio and is strong early game. Can take Scuttler and invade the enemy jungle.
But what about ganking?
Despite Irelia being pushed in she is not really in risk of being ganked since both junglers started bot. If Rammus ganks Irelia then Elise will probably counter gank (if the players are decent).
In Elise's case, she coud either try a gank mid or invade the enemy jungle. She can't really dive Nasus while not having at least an educated guess about where Rammus is. The best option? Invade the enemy jungle immediately. Ping for help if you need it.
What can you do with Lane priority?
"All lanes are pushing I can't do anything"
You probably have been in this position if you play jungle. Now you know that your lanes not only are pushing but you also have lane priority. You actually are in an advantageous situation.
How to play if you don't have Lane priority
As you see having Lane priority is way better than not having it. Play around it and plan your route accordingly.