Board width, usually measured as waist width, plays an important role in how the board works for a particular rider. Ideally the boot toe and heel are even with the board's edges. A little overhang is OK but too much and the toe or heel will dig into the snow when turning, greatly affecting control. If the toe and heel are too far in from the edge then getting the board onto it's edge becomes much more difficult, requiring excessive force from the rider.
The factors that seem to be most important are:-
1) How long have you been boarding?
Probably not very long, if you are asking this question. If you've been boarding a while, you may already know the answer. Anyway, beginners tend not to generate much of an angle between board and snow, and as such toe/heel overhang has minimal impact.
In addition, you may not know the answers to the questions below, and therefore maybe need to do some more riding.
2) What kind of boarding do you do (or want to do)?
If you love to carve the cordurouy more than anything else, then you'll probably be getting an alpine board. These are thin anyway, but you'll also be using higher angles, and hard boots (probably), however these speed hounds are looking for no overhang at all, so read on and substitute the minimal overhang for none.
If you are busting big powder fields 90% of your time, then overhang is almost irrelevant in powder, although the drag factor can become, well, a drag if it's crusty on top. Again, if you're at this stage of snowboarding, you should already know the width you want.
If you ride the resort (i.e. in bounds), do everything, and like doing carving with your freeride set up, or you like to ride the pipe/park, you need more information, read on.
3) What binding angles do you use?
If you run with high angles, then you have less lateral foot print. If you run low angles (+/- 15 degrees), then you have more chance of overhang.
4) Finally, how big are your feet? More importantly...
5) How big are your boots?
So here we are... the boots are important. For example, the Burton Work Boot of 1996 was a huge beast. However, the Burton Ruler Boot of 1997 was quite a small shell. Some boots have a bevelled toe and heel design, therefore moving the impact point away from the snow.
All of this leads down to the point that the answer isn't simple, and is different for each person. You should first decide on the best boots for you, for the type of bindings that you want to use. (It may be said that the boot which fits determines the bindings you use, not the other way round.)
Then ideally, get down do the shop, attach your bindings to the board of your choice, strap in, and then get up on the edge to the point that the toe edge of your boots touch the floor. Now do it heel side. Ensure that you put some down force onto the board, to ensure that the whole of the rail (the metal edge) is touching the floor. This will flex the board, potentially a lot, but don't be scared as it's supposed to do that!
If the board gets upto 60 degrees or so, and you have about 1cm (1/2 inch) overhang or less, then you're probably sorted.
If you have no overhang at all you may or may not have some problems, however this depends on the rider and the board.
If you have lots of overhang, and only make it to about 30 degrees or less, then either get a wider board, increase your binding angles, or don't expect to do any serious (fun) carving or make it up the wall of the pipe consistently.
If you are between 30 and 60 degrees, hmmmm, that overhang should be 1cm (1/2 inch) or less on both sides... check it out.
There was an extensive discussion on the newsgroup regarding the subject of needing a wide board, which got quite technical. Read it for yourself on google via this link, don't get your hopes up though, the results were inconclusive. :-)
Don't forget to check your heel edge. Too many people just talk and think about toe overhang. However, if the back of your bindings deck out quickly, you could find yourself on your arse sliding down a 45 degree slope after performing your first heelside turn, heading for those rocks/cliffs/trees. Believe me, you don't want to go there.
The jury is out on riser blocks. These are lumps of plastic that sit in between your board and your bindings, with the effect that your feet are now further away from the board, and allow you to lean over further. Some people use them and think they are great. Others use them and find that they lack a certain amount of control over the board. Others still just find them weird. You have to wonder though, if they're such a great idea, why don't people on normal width boards ride narrow boards with risers?
I know, I know, you've read this and you're thinking that it doesn't tell you anything, you want specifics.
If you have size 9's and use angles more than 12 degrees, you may be fine on a 248mm board. Less degrees than that, you'd better check.
If you have size 11 or over, and run angles of 15 or less, then you probably need to look for a wider board.
Personally, I ride with angles of 24/-6. I'm a freerider that likes to do all sorts. I do powder whenever it's possible, but when it's not, I may carve the groomed runs in the morning, and muck about in the pipe or park in the afternoon, or simply search harder for more powder. I have size 10 boots, use strap bindings and I ride a board with at least a width of 270mm. In the 2000/2001 season I switched to a 280mm board. I found that my riding had reached a level where I was occassionally decking out my boots on my 270mm board. I now have no overhang and I can carve as far over as I like. I don't worry about decking out and the board is fast and responsive - it is definitely not a problem getting edge to edge. This seems strange to other people, but at the end of the day, the most important thing is: It works for me.
I've heard of some people with 13's who claim they don't need a wide board. I'd say that they don't do any carving or minimal pipe work. Again though, it works for them, and presumably they're happy.