$[X,Y] = XY - YX$

See here.

We can "multiply" vector fields $X$ and $Y$ by considering them as differential operators (as, at each point they are Tangen vectors, but more precisely, vector fields can be seen as derivations on the space of smooth functions on a manifold), and so $XY$ means $X$ composed with $Y$.

Lie brackets can be seen as ways that one vector field (seen as an infinitesmial diffeomorphism) acts on another vector field. Basically, the Lie bracket gives a new vector field corresponding to the rate of change of the preimage of the vector $Y$ under the flow given by vector $X$. – So it says in which direction would I need to change vector field $Y$ at each point, so that the flow takes it to the original vector field $Y$