It means that you define quantities with respect to special reference frames according to the physical situation at hand. One example is the proper time, which is a scalar measure of time for a given particle, dependent on its trajectory. Physically it's defined as follows:
The trajectory is given in any inertial reference frame (let's call it the lab frame) by the four vector,
(x^{\mu})=\begin{pmatrix}t \\ \vec{x}(t) \end{pmatrix}.
Here, and in the following, I set the velocity of light, c=1 (natural units).
Now you define the infinitesimal proper-time increment \mathrm{d} \tau, as the time in a momentary inertial reference frame, with respect to which the particle is at rest. This you do at any time, t, and add up all these time increments. This defines proper time.
Now the quantitiy
\mathrm{d} x^{\mu} \mathrm{d} x_{\mu}=\mathrm{d} t^2-\mathrm{d} \vec{x}^2=\mathrm{d} t^2 [1-\vec{v}(t)^2] \quad \text{with} \quad \vec{v}(t)=\frac{\mathrm{d}}{\mathrm{d} t}\vec{x}(t)
is a Lorentz invariant. For a massive particle that's always a positive quantity, and thus you can take the square root of it. Since in the momentary rest frame of the particle, you have \vec{v}=0, the proper-time increment can be expressed entirely in terms of the lab-frame-time increment via
\mathrm{d} \tau= \mathrm{d}t \sqrt{[1-\vec{v}^2(t)]}.
"Adding" these increments up thus means to integrate from the initial time, t=0, to some lab-frame time t. Thus you get the proper time of the particle as a function of lab time by the integral,
\tau(t)=\int_0^t \mathrm{d} t' \sqrt{1-\vec{v}^2(t')}.
This is a strictly growing function, and thus you can use \tau as well as a measure of time as the lab time, t. The good thing is that the proper time, by construction, is a Lorentz scalar.
This gives you the opertunity to describe the motion of the particle in a manifestly Lorentz covariant by giving the lab-frame coordinates as function of proper rather than lab-frame time. Then you describe the motion as a trajectory in four-dimensional space time, x^{\mu}(\tau) with a scalar parameter, which is "natural" in the sense that it is physically determined by the situation at hand, namely the motion of a single particle.
Then the four-velocity is given by
u^{\mu}(\tau)=\frac{\mathrm{d}}{\mathrm{d} \tau} x^{\mu}(\tau),
which is a Lorentz-covariant four-vector as is x^{\mu} since the proper time, \tau is a scalar.
Of course, you always have
u^{\mu}(\tau) u_{\mu}(\tau)=\left (\frac{\mathrm{d} t}{\mathrm{d} \tau} \right)^2 - \left (\frac{\mathrm{d} \vec{x}}{\mathrm{d} \tau} \right)^2=1.
then the energy-momentum four vector of the particle is defined by
p^{\mu}=m u^{\mu},
where m is the proper (or invariant) mass of the particle, which is a scalar. Because of the previous equation, you have the relation
p_{\mu} p^{\mu}=E^2-\vec{p}^2=m^2 \; \Rightarrow \; E=\sqrt{\vec{p}^2+m^2}.
This manifestly covariant description is very useful to find Lorentz covariant equations of motion.
