First, make sure the image size (height and width in pixels) is as small as possible without detracting from the image. Then, its a question of saving the image in an appropriate format, with as small a file size as possible.
There are many file formats for digital images, but the 'universal' ones, read by all browsers, are .gif and .jpg. There is a third type, .png, but sadly, Microsoft Internet Explorer 6 doesn't support the alpha transparency feature, meaning that a .png graphic may not appear as the dsigner intended. However, good news! Internet Explorer 7 does support .png graphics and as this browser becomes more popular, cooler graphics utilising the alpha transparency will emerge!
The .gif format is best for logo's etc., basically anything that is not a photograph. To optimise a .gif means to reduce the number of colours needed to produce the image while retaining the quality at an acceptable level.
The .jpg format is best for photographs. In order to trade off image quality with file size, the 'quality' of the image can be degraded. For most images, instead of using 100% quality, 50% will give an almost identical image but with a smaller file size.