In guitar terms, a cutaway is a section of the body that has been removed from next to the neck. This is done to provide easier playing access to higher frets, but in acoustic guitars this comes at the cost of making for a 'smaller', 'weaker' sound. In electric guitars big cutaways can cause problems with getting the guitar to balance properly on a strap, although this is not a major issue.
A 'double cutaway' is where there are removed sections on the bass and treble sides of the neck. In single cutaway designs, the cutaway is almost always on the treble side of the neck. Gibson's Les Paul model, and Fender's Telecaster are examples of guitars with a 'single cutaway', and Gibson's SG and Fender's Stratocaster are typical of double cutaway guitars. Very rarely are double cutaways seen on acoustic guitars, but they are very common on electrics. This is primarily because a double cutaway design makes for a much weaker neck joint than a single cutaway design, and acoustic neck joints tend to be a little weaker than their electric counterparts. It's also relevant that not many acoustic guitars are designed with scorching lead guitar work in mind, so while upper fret access is useful, it's not really crucial.
In acoustic guitar design there are two basic types of single cutaway designs, the Venetian cutaway and the Florentine cutaway. The primary difference betweent the two is that the Florentine cutaway has a sharp point to the leftover 'body horn', where the Venetian cutaway is usually very rounded off. An example of a guitar with a florentine cutaway is the Gibson ES-135. Examples of a Venetian cutaway can be seen in the entire Taylor line of acoustic guitars.
Venetian cutaways tend to be more aesthetically pleasing, but the amount of time and work involved in convincing a piece of wood to bend itself around such a sharp angle is far more than the time required to simply cut the wood and reglue it on a sharp edge with some binding.
In solid body guitar designs the types of cutaways are many and varied, and pretty much anything is acceptable, as long as it doesn't throw the weight balance of the guitar out, and allows reasonable neck access.