You will advance to Level 2 during the Clash of Clans Tutorial
You can see your current level at the top left of the screen. This also shows your current amount of experience points. Levelling up allows you to recruit players into your clan more easily. You get experience, required for levelling up, by completing buildings. The more expensive and the longer a building takes to complete, the more experience you are rewarded with.
A small amount of experience is also given for removing rocks and trees, and destroying enemy Town Halls. Levels only really indicate how much someone has been playing Clash of Clans
and are not used to rank players in multiplayer.
Experience Points are earned for upgrading, fighting, gifting troops - generally any task in the game, if meaningful, will contribute Experience Points to your list.