The Adobe RGB (1998) colour space is an RGB (RED, GREEN BLUE) colour space developed by Adobe Systems, Inc. in 1998. It was designed to encompass most of the colours achievable on CMYK colour printers, but by using RGB primary colours on a device such as a computer display.
sRGB is the standard RGB colour space created cooperatively by HP and Microsoft in 1996 for use on monitors, printers and the Internet, and subsequently standardised by the IEC as IEC 61966-2-1:1999. It is often used as the "default" colour space for images that do not contain any colour space information, especially if the images are stored as 8-bit integers.
To better understand which one to use, you must first understand the difference between the two. AdobeRGB, by all accounts, is better, as it represents a wider range of colours. How much better? They say that AdobeRGB is able to represent about 35% more colour ranges than sRGB is able to. But does that make it the best for photography? Not exactly, as the world works with sRGB far more than it does with AdobeRGB.
How to Accurately Convert Your Photos from AdobeRGB to sRGB
If you use a tandem of Lightroom and Photoshop, Adobe makes this conversion process painless for you. My workflow and many others consist of loading images into Lightroom, making basic corrections, then importing the image directly into Photoshop. Upon importing to Photoshop, you can have your images converted for the web with just a few simple setting adjustments. Simply go to Edit>>Preferences>>External Editing and adjust your colour space to sRGB when being imported into Photoshop. This technique is the most preferred, as it'll automatically convert all images you export to Photoshop to sRGB, without any colour loss in the web format. This will also allow you to keep both an AdobeRGB copy of the image for print, and an sRGB version to use for the web and everything else.
If you work without Lightroom and still want the benefits of AdobeRGB colour space, you can also convert your images for the web in Photoshop. Simply navigate through your menus to Edit>>Convert To Profile and change your destination space to sRGB after editing your image. To ensure that you do this every time, I recommend you incorporate it an action used for saving your images. Remember, failure to convert your images prior to saving them for the web will result in dull and unflattering colour tones.
All of this is very confusing and leaves you feeling overwhelmed, you can switch your camera to sRGB colour space. It'll still allow you to photograph and print beautiful images. However, if you're shooting specifically for print, AdobeRGB offers more range and versatility in the images taken. It all really comes down to personal preference, AdobeRGB does offer more colours, but at the cost of complicating things for a subtle difference in your photos. However, if you're a perfectionist, like me, the extra steps taken to shoot in AdobeRGB may be worth the headache to achieve nicer prints, and get the best of both worlds.