Quadrature modulators are used to conserve bandwidth for a given data rate. This is accomplished by modulating two orthogonal data streams onto a common carrier. If the phases and amplitudes of both data stream (inphase "I" and quadrature "Q"), then one of the sidebands is completely cancelled out. If there is no DC bias feedthrough, then the carrier itself is completely cancelled out. In practice, complete cancellation is never accomplished, but without too much work, achieving 40 dB of sideband cancellation is not hard to do. Even 60 dB is relatively easy; however, preventing drift due to thermal and mechanical effects is not so easy, and the result is that a "textbook" quadrature alignment during alignment can look pretty bad over time.
Without showing all the trigonometry and algebra steps inbetween, here are the basics of quadrature modulation. See the chart below.
InPhase Data (I)  Quadrature Data (Q)  Carrier (C)  
After Multiplying I & Q by the Carrier and Adding  
Note that the D cos(Ω_{c}t) term is the carrier and disappears if D (the DC component) equals zero. 

Lower Sideband Envelope (LSB)  Upper Sideband Envelope (USB)  
where G is the amplitude imbalance (ratio) and f is the phase imbalance, and 

Sideband Suppression Equation  
Phase Error (f)  
Picture of Sideband & Carrier Suppression  
Chart of Sideband Suppression as a Function of Phase and Amplitude Imbalance  
This chart was created with Excel, using the above equations. 
Related Pages on RF Cafe
 Amplitude
Modulation
 Frequency Modulation

Quadrature (I/Q) Modulator Sideband Suppression

Bessel Functions & Graphs

Modulation Principles, AM Modulation,
NEETS
 Modulation Principles,
FM Modulation, NEETS
 Modulation
Principles, Demodulation, NEETS
 Frequency Mixer, Converter, Multiplier,
Modulator Vendors