What is a guitar cutaway (with pictures)

A guitar cutaway is when a guitar's body is "cut away," like this:

Cutaways make it easier to play lead guitar licks and melodies up higher on the neck, like above the 12th fret. 

Some acoustic guitars have cutaways too, but certainly not all.* Similarly, not all electric guitars have cutaways.

Some guitars, like the Fender Stratocaster, have a double cutaway:

How do cutaways affect the sound of a guitar?

I think that this is one of those questions that raises a bit of controversy in guitar-manufacturing circles. Since it is a tough question to answer, I'll just answer “definitely, maybe, kinda, I’m not sure, hmmm, I think so, I think not.” Seriously, it really depends on who you ask.

If you want an expert opinion, contact a couple of your local guitar shop owners and ask them to talk shop with you. If you're after a superbly informed answer, contact some guitar manufacturers and chat them up. 

One final thing to consider: Jimi Hendrix would have sounded like Jimi Hendrix no matter the guitar he played. In my opinion, sound has more to do with the guitarist than it will ever have to do with whether or not the guitar has a cutaway.

*All acoustic guitar manufacturers know that they must design guitars that have cutaways as well as non-cutaway guitars too.