To explain exactly what this means, it may be a bit easier to first explain the concept of "Galilei-invariance" since this is a little simpler mathematically. Here is the Galilei transformation for transforming between coordinates of different inertial reference frames in Newtonian physics:
x'=x - vt
y'=y
z'=z
t'=t
x=x' + vt'
y=y'
z=z'
t=t'
To say a certain physical equation is "Galilei-invariant" just means the form of the equation is unchanged if you make these substitutions. For example, suppose at time t you have a mass m1 at position (x1, y1, z1) and another mass m2 at position (x2, y2, z2) in your reference frame. Then the Newtonian equation for the gravitational force between them would be:
F = Gm1m2/[(x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2]
Now, suppose we want to transform into a new coordinate system moving at
velocity v with respect to the first one. In this coordinate system, at
time t' the mass m1 has coordinates (x1', y1', z1') and the mass m2 has
coordinates (x2', y2', z2'). Using the Galilei transformations, we can
figure how the force would look in this new coordinate system, by
substituting in x1 = x1' + vt', x2 = x2' + vt', y1 = y1', y2 = y2', and
so forth. With these substitutions, the above equation becomes:
F = Gm1m2/[(x1' + vt' - (x2' + vt'))^2 + (y1' - y2')^2 + (z1' - z2')^2]
and you can see that this simplifies to:
F = Gm1m2/[(x1' - x2')^2 + (y1' - y2')^2 + (z1' - z2')^2]
In other words, the equation has exactly the same form in both coordinate systems. This is what it means to be "Galilei invariant". More generally, if you have any physical equation which computes some quantity (say, force) as a function of various space and time coordinates, like f(x,y,z,t) [of course it may have more than one of each coordinate, like the x1 and x2 above, and it may be a function of additional variables as well, like m1 and m2 above] then for this equation to be "Galilei invariant", it must satisfy:
f(x'+vt,y',z',t') = f(x',y',z',t')
...for all possible values of v.
From this, it's pretty simple to see what it must mean for a given physical equation to be "Lorentz invariant" as well. Here are the Lorentz transformation equations in three dimensions:
x'=gamma*(x - vt)
y'=y
z'=z
t'=gamma*(t - v*x/c^2)
x=gamma*(x' + v*t')
y=y'
z=z'
t=gamma*(t' + v*x'/c^2)
So, if you have some physical equation f(x,y,z,t), then for it to be "Lorentz-invariant" it just must have the following property:
f(gamma*(x'+v*t'),y',z',gamma*(t'+v*x'/c^2)) = f(x',y',z',t')
...for every v<c.
This is just a mathematical property of a given equation or set of equations, it is simply a matter of calculation to check if the equation satisfies it (the equation for Newtonian gravity would not have this property, so it would not be Lorentz-invariant). Maxwell's laws have this property of Lorentz-invariance, as do all the most fundamental laws currently known (such as the laws of quantum field theory).
