The speed that we use in relativity is the four-velocity. Just as ordinary velocity is a 3D vector that describes the rate of change of your position in space with time, the four-velocity is a 4D vector that describes the rate of change of your position in spacetime with proper time. This four-velocity is an invarient and all observers in all inertial frames will agree on its value.
Proper time is a concept that doesn't exist in non-relativistic physics. You'll be familiar with the fact that clocks on fast moving spaceships run slow due to time dilation, and this makes defining time a bit tricky. Your proper time is the time shown on a clock you are carrying i.e. the time shown by a clock that is stationary relative to you. In special relativity the proper time is an invariant and all observers in all frames will agree on its value. This makes it a very useful concept, and if you study relativity you'll run into proper time all over the place.
Anyhow, in 3D space where your position is given by a vector $(x, y, z)$, the velocity is given by:
$$ \vec{v} = \left(\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}\right) $$
In 4D spacetime we choose an inertial frame using coordinates $x$, $y$, $z$ and $t$, and the position some object is then given by the four-vector $(t, x, y, z)$. The four-velocity is then given by:
$$ \vec{U} = \left(\frac{cdt}{d\tau}, \frac{dx}{d\tau}, \frac{dy}{d\tau}, \frac{dz}{d\tau}\right) $$
There are a few interesting things to note about the four-velocity. Suppose you are measuring your four-velocity in your own inertial frame. In this frame you are stationary so $dx = dy = dz = 0$, and the time $t$ is the same as the proper time $\tau$ (because that's how proper time is defined). So your four-velocity is:
$$ \vec{U} = \left(\frac{cd\tau}{d\tau}, 0, 0, 0\right) = \left(c, 0, 0, 0\right) $$
So even when you're standing still your four-velocity is $c$. That's because even you you're not moving in space you are moving in time, and in fact you're moving though time at the speed of light.
