The Wikipedia article on indicated airspeed has a good description. An airspeed indicator is actually more of a "dynamic-pressure" meter, with the dynamic pressure converted to airspeed.
Dynamic pressure is $q=\frac{1}{2}\rho V^2$ where $\rho$ is density and $V$ is airspeed.
The Wikipedia article on the lift coefficient explains that lift is proportional to dynamic pressure, the area of the wings, and the lift coefficient, which in simplified terms can be considered a function of the angle of attack.
$L=\frac{1}{2}\rho V^2 S C_L(\alpha) = qS C_L(\alpha) $
The takeoff speed is the speed at which you will have enough lift to get the airplane off the ground at the angle of attack that the airplane will have post-rotation. So, for a given aircraft at a given weight, wing area, post-rotation angle-of-attack, and lift curve, you will be able to take off at a particular dynamic pressure.
In fact, all the aerodynamic forces on the aircraft are proportional to the dynamic pressure. So that's why stall speed, never-exceed-speed (above which the aerodynamic forces could start damaging the aircraft structure), etc. are all given in indicated airspeed - because it's a proxy for dynamic pressure.
True airspeed is the speed of the aircraft relative to the air mass it's flying through and can be calculated from indicated airspeed by correcting for density and temperature. Your ground speed is then the true airspeed added to the wind speed.