What does "leveling up" mean?
What does "leveling up" mean? And why not spell it "levelling"? Even then it doesn't make any sense.
A "level" is the number that represents your overall skill. The higher the number, the more skilled you are in that language. It all comes from MMO games like Dungeons & Dragons or World of Warcraft, in which your character gains levels in order to become stronger. It doesn't really have any use and it's there merely to compare your progress to others and motivate you to go further until you reach 25, which is the maximum level one can obtain, as louis.vang explained.
You gain levels by earning xp. XP stands for Experience Points. You earn experience points by doing excercise and strenghtening your skills. When you collect enough xp, you gain a level, for example:
Level 2 - 20 XP Level 3 - 50 XP Level 4 - 100 XP etc.
This is NOT the exact number of XP you need to get these levels on Duolingo, I'm just giving an example. If you want to see how many xp (experience points) you need to get to your next level, open your Profile Page (hover over your name in the upper right, then click Your Profile) and it will say on the right. Hope this helps.