With a lateral load, a building is usually displaced by a little bit; this can be a matter of, for discussion, say an inch. This is the 'delta'
The entire vertical load caused by the weight of the building and the occupants, snow, equipment,etc. is now applied to the supports by this offset of an inch. This is the 'P'.
The entire weight times the one inch displacement is considered as an additional 'overturning moment'. This is the 'P-delta'.
This overturning moment has the same effect as increasing the lateral load by an amount 'P-delta' divided by the storey height.
You might remember (from school) the derivation of the differential equation for a column under an axial load. When the column deflects (Delta-1), the axial load (P) causes a secondary moment (PxDelta-1), which in turns causes additional deflection (Delta-2), which causes additional moment, etc.. The differential equation is the solution to this problem.
dik's answer is also correct with regard to a system of members. It is basically, additional loads on members (or the system) due to deflection from the applied loads.
If you do not apply the P-Delta condition to your member or system, you are required to determine the "moment magnification" of each member under axial load. The moment magnification equation(s) are derived from that differential equation mentioned above. Depending upon the characteristics of the member and the loading, the magnification ranges from negligible to significant.
For column design, there are actually two PDelta effects to consider.
First - there is the frame sidesway condition that dik and rowe discuss above.
Second - the individual column may deflect along its length due to frame deflection or lateral loads applied directly to the column. Think of an "S" shaped deformed column in frame.
Now along the length of the column there is axial compression that can add additional, secondary moments to the column itself. This is also a PDelta effect that must be considered.
An analogy would be taking a bow (from a bow and arrow) and setting one end on the ground, and the other straight above. Due to the curvature of the bow, if you push down, the bow bends even more.
