Violin sizes and length
The size of all members of the violin family is described by a single letter from 'A' (smallest) to 'G' (biggest). The letter groupings refer to the half-size difference between each sound-producing section. For example, a 4/4 violin has a body that is four-halves the length of a full-sized instrument; similarly, a 3/4 violin's body length is three-fourths that of full size and so on.
An instrument smaller than 1/16 sounds strange and probably isn't built correctly. If you can get your hands on one it will likely be bent out of shape by its new owner within 30 seconds of playing it for the first time.
The smallest instruments were once made for the young and very small. However, these "toys" didn't survive into modern times because of their lack of volume and tonal quality. Most musicians prefer to play the violin that is comfortable, not one that requires them to curl up on top of it like a contortionist just to reach the fingerboard.
While the physical size remains the same, there are some important differences between models larger than 1/8 or so: especially following the change from gut strings (where string tension was low) to steel strings (which exert higher tension). Players generally prefer instruments with slightly thicker tops and backs made from more dense spruce as they age in order to maintain a strong tone.
1/10 Violins
1/10 violins are tiny and very uncommon. It is nearly impossible to play one without an A-string that is so thin it buzzes terribly if you try to play on the A string at the first position (where your second finger on a down bow is also touching the E or higher string). They make a decent wall decoration but don't expect them to sound good.
1/8 Violin
1/8 violin sizes have been largely replaced by 1/10s as they were made popular in the past century by Suzuki methods of teaching children, as opposed to learning piano or recorder as some other methods encouraged. They still show up from time to time though, notably those built by German makers such as Kemal Gekich.
1/8 violins have a large body and so are sized between 1/10s and 3/4s. They usually sound best in the C-string tuning (C, G, D, A), though if you push your teacher to let you tune up to E or even F there might be string issues at first.
1/4 Violin
1/4 violins are one of the most popular sizes from Suzuki method programs and other beginning music stores for children learning stringed instruments because it is uncomplicated and easy to play in a lot of situations. A larger instrument might look impressive, but a smaller size makes sense when you have small hands.
A 1/4 violin is rarely used by professionals as they are too small for all ten fingers to fit on the fingerboard comfortably, even if you push down hard on the strings and your teacher can't hear how much you squeak and squawk through the first position.
1/4 violins, however, are great for beginning students to play on if they have a lot of energy and drive to practice their bowing skills.
1/2, 3/4, and 7/8 Violins
1/2, 3/4, and 7/8 violins are the most common sizes you will see in any store that sells brands such as Yamaha or Cecilio. They're also made by smaller luthiers such as Gallatin Valley Strings. Anything larger than a 1/2 is pretty rare nowadays, though some people still like to play on them due to their big size and loud projection of sound.
The C string on a 3/4 violin can be tuned up to D if you wish (just like with a 1/4) and it will sound better than usual since there isn't an extra-thick wound E string covering it up. The A string can be tuned up to B without sounding too terrible if you can't get your hands on a 1/2 or 1/4 sized instrument.
7/8 violins are very popular for professional musicians who have big enough hands to reach all the positions comfortably but wish to play on an instrument that projects more sound. These are difficult instruments to find nowadays, though they still pop up every once in a while in internet auctions and rare violin shops.
4/4 Violin
4/4 violins are the standard size for professional musicians. You can find these almost anywhere - even big box stores like Wal-Mart sell them under their house brand names of Cecilio, Pure, and whatever else they call their cheap violins these days.
A 4/4 violin is sized perfectly for an adult's hands to fit on comfortably with each finger in its own place (provided you've practiced enough to get your fingers in that position).
Players with larger hands may prefer a 7/8 or fatter body. Players with smaller hands often prefer a 1/2, 3/4, or 5/8 violin because it will feel easier and lighter in weight.
What is so special in 4/4 violin?
Since it is such a common size, most music stores will have 4/4 sized violins in stock. It's also set up to be tuned perfectly GDAE or violin tuning if you prefer that instead of 3rd position (G tuning). 4/4 violins are sized by luthiers to be proportioned just right - not too big and not too small.
Of course, if you're a child learning the violin you may prefer 1/8 or even 1/10 sizing so that your hands can fit on the fingerboard more easily.
If you're an adult who has never played before but wants to learn, 4/4 is probably what you should ask for when shopping at music stores. It's good enough for professionals all over the world (yes, anyone wearing a tuxedo who isn't playing pop songs), it won't cost as much as buying another size, and there will always be music written for it because almost every instrument has 4 strings (including Kazakh dombra, Finnish jouhikko, Indian sitar, and Chinese erhu).
Sound of 4/4 violin
4/4 violin is the standard for professional orchestras, so expect to have a lot of tone coming out of it. String players who play the 2nd violin, viola, or cello should be able to match the sound decibel level of 1st position if you are playing with other people who are also using 4/4 size instruments.
Don't be surprised if you use up all the rosin on your bow every 20 minutes when starting out - this is normal due to having so much more surface area hitting the strings compared to smaller sizes! You may want to start off by buying a cheap bow that isn't too expensive before investing in a nice bow until you know whether or not you're going to stick with playing.
No matter what size you get, be sure to keep it in good shape by rosining the bow every time you play and keeping it in its case when not practicing or performing. It's a good idea to clean the rosin off the strings after every use too so that they don't get frayed and break more easily than they need to.
Final word
The 4/4 violin is the standard size in the professional orchestra (we mean, it's what Mozart used). If you can't afford a good instrument right away, getting the 4/4 will let you get years of use out of something that should be at least decent quality. This isn't true for instruments smaller than 1/2 or larger than 7/8 since they are harder to find and more expensive - if possible, only get these sizes if you want to invest more money into them.
For an adult beginner who has never played before but wants to give it a try, pick up a 4/4 so that your fingers have space on the fingerboard. You don't need anything fancy until know you enjoy playing enough to practice daily - the first violin you buy will only be good for beginners, so it doesn't have to hold up through your entire life as a player.