sRGB is the most standard, and is best used when you want to show images on the web or to send images to a printing lab (unless the lab specifially states that they will accept Adobe RGB). Web browsers understand what sRGB is, and can display the image correctly.

Adobe RGB is more advanced - it has a larger colour gamut, and so can display more colours. The advantage of this means that with more colours you can get more detail, smoother colour gradients and so forth. The drawback of this is that your image editor must recognise Adobe RGB (and many do now). If you post an Adobe RGB tagged image to the web, you will notice it will be displayed darker than usual - this is because the web browser cannot process and display all the colours. It just treats it as a standard sRGB image, and the colour mapping becomes confused. For your own editing and printing, it makes sense to use Adobe RGB (especially now that many advanced digital cameras allow you to capture images with an sRGB or Adobe RGB colour space - they used to only be sRGB).