Here's one version of the Lorentz transformation equations, which covers the case where the primed frame is moving at velocity v along the x-axis of the unprimed frame:
x' = \gamma (x - vt)
y' = y
z' = z
t' = \gamma (t - vx/c^2)
where \gamma = 1/\sqrt{1 - v^2/c^2}
and
x = \gamma (x' + vt')
y = y'
z = z'
t = \gamma (t' + vx'/c^2)
So what Lorentz-invariance means is that if you take some equation for a law of physics written in terms of x,y,z,t coordinates and use the Lorentz transform to make substitutions and rewrite this equation in terms of x',y',z',t' coordinates, the equation will end up looking the same as if you had just replaced x with x', y with y', z with z' and t with t'--the equation should have exactly the same form in both coordinate systems.
Here's an example, involving "Galilei-invariance" rather than Lorentz-invariance because the math is simpler. The Galilei transform for transforming between different frames in Newtonian mechanics looks like this:
x' = x - vt
y' = y
z' = z
t' = t
and
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 m_1 at position (x_1 , y_1 , z_1) and another mass m_2 at position (x_2 , y_2 , z_2 ) in your reference frame. Then the Newtonian equation for the gravitational force between them would be:
F = \frac{G m_1 m_2}{(x_1 - x_2 )^2 + (y_1 - y_2 )^2 + (z_1 - z_2 )^2}
Now, suppose we want to transform into a new coordinate system moving at velocity v along the x-axis of the first one. In this coordinate system, at time t' the mass m_1 has coordinates (x'_1 , y'_1 , z'_1) and the mass m_2 has coordinates (x'_2 , y'_2 , z'_2 ). Using the Galilei transformation, we can figure how the force would look in this new coordinate system, by substituting in x_1 = x'_1 + v t', x_2 = x'_2 + v t', y_1 = y'_1, y_2 = y'_2, and so forth. With these substitutions, the above equation becomes:
F = \frac{G m_1 m_2 }{(x'_1 + vt' - (x'_2 + vt'))^2 + (y'_1 - y'_2 )^2 + (z'_1 - z'_2 )^2}
and you can see that this simplifies to:
F = \frac{G m_1 m_2 }{(x'_1 - x'_2 )^2 + (y'_1 - y'_2 )^2 + (z'_1 - z'_2 )^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 x_1 and x_2 above, and it may be a function of additional variables as well, like m_1 and m_2 above] then for this equation to be "Galilei invariant", it must satisfy:
f(x'+vt',y',z',t') = f(x',y',z',t')
So in the same way, an equation that's Lorentz-invariant should satisfy:
f( \gamma (x' + vt' ), y' , z', \gamma (t' + vx' /c^2 ) ) = f(x' ,y' ,z' , t')