Relativistic invariance (or Lorentz invariance) means "the same regardless of frame of reference". For example, a relativistic invariant quantity would be the same if you measured it while you are at rest versus if you measured it while you are moving at a constant velocity. If it is a quantity requiring the Lorentz transform when shifting frames of reference, then the value of the quantity is unchanged by the transform, and often the phrase relativistic invariance is used specifically to mean that particular usage.
An example is the speed of light (c), and the discovery of its invariance (through measurement) was a prime motivator for the development of relativity and the Lorentz transform. Applying the transform to the speed of light does indeed always yield the same value. Mass (as the word is generally used, i.e., as the "rest mass") is invariant, but only because, as defined, one doesn't apply the Lorentz transform.
The terms are also used for laws of physics, i.e., equations, that remain true in different frames of reference. Some familiar laws are invariant, and some have relativistic versions that are invariant, i.e., that make their point using only invariant quantities, and some are "meta", e.g., the laws of relativity itself.
Lengths and time intervals are famously not invariant: a foot-long ruler no longer measures to exactly a foot when measured from a frame of reference in which the ruler is moving. A spacetime interval (for two events happening some distance and time interval apart, it is square root of the difference between the distance squared and the time interval squared) does remain unchanged under the Lorentz transform. The minimum time interval or minimum distance between two events (among all frames of reference) is also a fixed number, but that's because such a minimum would occur in a specific frame of reference: its lack of variation is because we've stepped back from comparing multiple frames and are commenting on a property of a single frame that is specific to that distance or interval.