Offset not only affects how...
Offset not only affects how far the wheel/tire sticks out of the fender well but also its proximity to suspension components like knuckles, and control arms.
More about offset
With clearances measured, you're almost ready to look really smart in front of your friends. First: A wider tire on a similarly offset wheel reduces the gap the same amount on both sides but adding 1mm of offset moves the wheel away from the fender-closer to the inner wheelwell-by 1mm. Usually, the inner and outer gaps will be different and this is why new wheels require different offsets. To center the tire so that the inner and outer gaps are equal, take the distance of the outer gap, subtract the inner gap from that figure, and divide this number by two. Add this figure to the old wheel's offset, this gives you the new wheel's offset. For example, imagine a stock 195mm tire and a new 225mm one. Picture inner and outer gaps, 30mm and 21mm, respectively. Give the old wheel a 45mm offset but first be sure the new tire will fit. Since the total gap in this example is 51mm (30mm plus 21mm), and the tire width increased by 30mm, then there is 21mm left over. Even after assuming that the new tire might be an extra 1/4-inch (6.4mm) wider than expected, there's still more than 14mm clearance. In short, 225mm will work. Subtract the inner gap from the outer gap, which is -9mm (21mm minus 30mm), to obtain the new wheel's offset. Take half of this number (-4.5mm or simply round to -5mm), add this figure to the old offset, which was 45mm to net your 40mm offset. A quick check of Volk TE37 and CE28 wheel offsets reveals a +35mm and +42mm offset available in 15x7, both of which will work just fine.
On a related note, you also need to know about backspacing. Specifically, it's the distance between the wheel's inboard outer edge and its hub-mounting surface. Proper backspacing calculations ensure against any unwanted clearance issues between tires and suspension members, especially when larger wheels and tires are used. Enough said.
All you need to properly measure...
All you need to properly measure offset is a tape measure and a straight edge. We're measuring the wheel's backspacing here, which is needed to calculate its offset.
How to measure it
Offset can be measured with or without the tire installed. When measuring with the tire, offset can still be measured without deflating it, however, it might be more accurate to measure offset on a bare wheel but that doesn't mean you can't get within a millimeter or two of the true figure with tires in place. After all, as long as offset is measured within a 2mm tolerance, things should be fine. Should you go the tire-installed route, it's easiest to measure the overall tire width using a straightedge placed across the tire to a level spot on the ground. Next, measure from the same straightedge location to the wheel's mounting surface. The only tools required to measure offset are a long straightedge, a tape measure, and this simple mathematical formula:
Offset equals 0.5 (tire or wheel width) minus the distance from the mounting surface to the tire or rim edge.
Offset and track widths
Altering offsets also changes track widths. Reduced offsets result in larger track widths. This can improve cornering speeds by reducing lateral load transfer from the inside tires to the outside tires. By keeping the tire loads uniform, tires can generate increased lateral grip. This is why most race car tires are as far outboard as possible. On the other hand, changing the wheel width does not change the track.
Another way to increase track without having to purchase new wheels is to install wheel spacers. Spacers are available in different thicknesses, from about 5mm to 30mm or more, depending on the application. Eibach manufactures one of the nicest sets of spacers on the market. Their spacers are precision machined with tight tolerances, which helps ensure that wheels stay securely attached to their hubs. Eibach manufactures 5mm non-hub centric and 15mm hub centric options, each with longer wheel studs.