As David Bridgen said, a x1 probe is just a piece of screened wire, but it is fairly special wire, it's of very low capacitance (which is important).
A x10 probe includes a 9M resistor in series with the inner core, with a trimmer capacitor across it. The 9M resistor acts as a potential divider in conjunction with the 1M input impedance of the scope. The trimmer is adjusted to compensate for the input capacitance of the scope (including the probes cable) - whereas the resistor has to be 9x the input impedance, the capacitance has to be 1/9th of it's input capacitance.
Scopes commonly (always?) have a probe adjustment terminal on the front, you set the probe to x10 (if it's switchable) and clip it to the terminal. You 'should' get a nice squarewave on the scope, you adjust the trimmer capacitor until you do - it's VERY easy to do, and totally obvious where the correct point is.
Incidently, adjusting the capacitor is essentially a 'treble' control, just like on your HiFi - and gives exactly the same effect, which is why squarewaves are often used for testing amplifiers.