I have started analyzing my games in Lichess. I'm wondering what does
Depth mean in the Stockfish analysis?
It means how many half moves ahead stockfish looked from the current position. It doesn't mean it looked at all the moves possible to that extent thought, it trims what it analises, otherwise it would be unable to reach such depths, this means a mate in 10 for example may only be found when stockfish searches in greater depths. In general, greater depths mean better analysis and moves suggested by stockfish. The only limiting factor is time and processing power.
Depth counter shows the latest status of the algorithm in the search tree. Depth 10 means the algorithm checked all moves in 10 node distance. Each node is a position (each half move changes the position, so depth of 10 translates to 5 moves).
As you wait longer and longer it will get deeper thus will provide better evaluation.