To add to Johannes's succinct answer: there are two other things about time that we believe to be absolute:

1. The topology of any web of causal links between events in space time and
2. The direction in time of causal links.

If we postulate that a cause-effect relationship can only propagate at a maximum speed of $c$, then if a cause comes before an effect in one reference frame, it comes before the effect in all reference frames. Geometrically, a vector $\vec{AB}$ drawn from an event $A$ such the head $B$ is inside the forward (or backward) pointing light cone with apex at $A$ is transformed by any orthochronous[1] Lorentz transformation such that the image of $B$ is still confined to the same light cone.

Indeed this property of the Lorentz transformation is what leads us to postulate that $c$ is a speed limit. If we heed this speed limit, then our everyday notion of causality - that causes always come before effects - is unchanged. This is one of the most remarkable things about relativity: even though proper times of different paths through spacetime between points can be shrunken or stretched a bit in twin paradoxes and so forth, most of our everyday intuitions about time - namely those to do with causality - are altogether unchanged by the notion of relative time that relativity describes.

[1]. An "orthochronous" Lorentz transformation is one that is the composition of any finite number of finite rapidity boosts (transformations between inertial frames of reference) and rotations.

To add to Johannes's succinct answer: there are two other things about time that we believe to be absolute:

1. The topology of any web of causal links between events in space time and
2. The direction in time of causal links.

If we postulate that a cause-effect relationship can only propagate at a maximum speed of $c$, then if a cause comes before an effect in one reference frame, it comes before the effect in all reference frames. Geometrically, a vector $\vec{AB}$ drawn from an event $A$ such the head $B$ is inside the forward (or backward) pointing light cone with apex at $A$ is transformed by any orthochronous[1] Lorentz transformation such that the image of $B$ is still confined to the same light cone.

Indeed this property of the Lorentz transformation is what leads us to postulate that $c$ is a speed limit. If we heed this speed limit, then our everyday notion of causality - that causes always come before effects - is unchanged. This is one of the most remarkable things about relativity: even though proper times of different paths through spacetime between points can be shrunken or stretched a bit in twin paradoxes and so forth, most of our everyday intuitions about time - namely those to do with causality - are altogether unchanged by the notion of relative time that relativity describes.

[1]. An "orthochronous" Lorentz transformation is one that is the composition of any finite number of finite rapidity boosts (transformations between inertial frames of reference) and rotations.