One can see some initial answers in Huybrechts' book "Fourier-Mukai Transforms in Algebraic Geometry"

1. ([Huybrechts], Prop 4.1) If two smooth projective varieties are derived equivalent, then they have the same dimension.

2. ([Huybrechts], Prop 3.10) If $X$ is Noetherian, then $X$ is connected if and only if $D^b(Coh X)$ is indecomposable.

I heard about the first part of the following argument as a folklore, someone should correct me if I am wrong:

3. The compact objects in $D^b(Coh X)$ are precisely those quasi-isomorphic to bounded complex of locally free sheaves, so $X$ is regular if and only if every object in $D^b(Coh X)$ is compact.

One can see some initial answers in Huybrechts' book "Fourier-Mukai Transforms in Algebraic Geometry"

1. ([Huybrechts], Prop 4.1) If two smooth projective varieties are derived equivalent, then they have the same dimension.

2. ([Huybrechts], Prop 3.10) If $X$ is Noetherian, then $X$ is connected if and only if $D^b(Coh X)$ is indecomposable.

I heard about the first part of the following argument as a folklore, someone should correct me if I am wrong:

(EDIT: The following is WRONG. However, there is some interesting discussion in the comments to this answer.)

3. The compact objects in $D^b(Coh X)$ are precisely those quasi-isomorphic to bounded complex of locally free sheaves, so $X$ is regular if and only if every object in $D^b(Coh X)$ is compact.