When you learn on Duolingo, you earn points and "level up." The XP levels you get in each language course correspond directly to how many XP (points) you earned while doing that course. You earn points by practicing and successfully completing lessons.
XP levels do not correspond to language proficiency or skill, but it does reflect effort and time spent. The Fluency Score, however, does reflect skill according how you have been performing on Duolingo. Learn more about that here.
Levels are displayed on your profile and near your avatar in forum discussions, next to the flag icon of the language you are learning.
The maximum level you can reach is 25.
How to see your level
To see your level on the website, go to "Your Profile", under your avatar:
Your levels will show up on the column on the right hand side, under "languages". You can also see your total points and how far you have to go to reach the next level.