Offset is one of a number of measurements applicable to a road wheel and it describes the distance from the centre line of the wheel which the mounting face of the hub is.
As a practical example, lets consider a 6J wheel (a wheel which is 6 inches wide). Offset is usually quoted as "ET" followed by an optional mathematical sign (+/-) and a number. A wheel with an offset of ET0 is described as a zero offset wheel. This therefore means that our 6J wheel will have an equal 3" of wheel sticking out infront of the hub and 3" of wheel sitting back over the brake disc. Offset is quoted in millimeters (mm) so for ease of conversion, a wheel with an offset of ET+25 will effectively be recessed into the arch by ~1" so will protrude 2" with 4" over the brake disc. Conversely a wheel with an offset of ET-25 (negative offset) will be pushed out closer to the wheel arch so will have 4" of wheel protruding from the hub and only 2" over the brake disc.
There are a number of reasons why you may wish to deviate from manufacturers standard offset. You may wish to fit wider wheels and encounter clearance issues with the rear face of the wheel touching the suspension leg at standard offset.
You may, as stated wish to give the car a more aggressive look so that the wheels and tyres appear to "fill" the arches better.
The effect on handling of deviation from manufacturers offset is to increase effective levels of grip whilst sacrificing a certain level of feel. More performance variants of cars employ something called "wide track". Track refers to the width between the wheels measured across an axle. For example, the Mk3 Golf VR6 used different front wishbones and ball joints to effectively push the wheels further out into the arches than the standard car.
What a wider track does is mean that cornering forces can be more effectively coped with by the outside tyres. Consider a push bike, very unstable and easy to push over. Add stabilizers to the bike and it now becomes harder to push over. Imagine increasing the width from the bike that these stabilizers extend and you will see how it becomes harder and harder to push over.
The trade-off however is that when a standard road car approaches the limits of traction it is very communicative to the driver through the steering feel that it's about to loose grip and progressively break away from it's intended line. If the driver presses on they may unsettle and ultimately loose control of the car. With a wide track car the limits are approached at a higher speed and the amount of information fed back to the driver is a lot shorter before the car breaks traction completely. Consider how many Porsche 911's are damaged on wet roads because they have such a high level of grip but it goes away very very quickly. Solutions to this immediate loss of traction may include the use of bias ply tires which always given more warning and allow the driver ample opportunity to scrub the tires well before traction is gone completely.
Related slightly to this answer is the fact that certain tuners use spacers to increase the track of the car which has the same nett effect as altering wheel offset as it pushes the wheel out by the width of the spacer from the hub.
One final note is that running a wide track through offset or spacers typically accelerates wear on semi floating wheel bearings. Full floating axles tend to favour larger negative offset.