Bicycle racing, at the professional level, is dominated by relatively small slender men. Cycling is a primarily aerobic sport, and extra mass, even muscle mass, is just more weight to carry over those hills.
But commuter cyclists come in all shapes and sizes, from flyweights who could (and maybe do) race at a competitive level to those who are often referred to as "Clydesdales," riders over 200 lbs. Personally, at 6'3" and well over 200 lbs I fall pretty firmly into that latter category, which makes me something of an expert in setting up bikes for big riders.
Except in the case of ultralight racing gear, the choices big guys (and girls) make aren't too different from those made by more average-sized riders, but there are a few things that are worth knowing or looking into.
First off, bicycles rarely have a posted weight limit, except in three cases. The first are lightweight racing-style bikes, which sometimes have a limit of around 180 lbs (this would be huge for a professional bicycle racer). The second case is folding bicycles, which have complicated frames with folding joints and long, unsupported seatposts. Folding bike weight limits are often in the 220 lbs range. The final category of bikes with posted limits are cargo bikes, that are often described as being able to carry x lbs of combined rider and cargo weight. These limits are often upwards of 400 lbs, and shouldn't concern any but the very largest of cyclists.
Other bikes don't have a posted limit, and if pressed, most manufacturers will usually hedge by saying something like "riders over 250 lbs should use caution" or "they're fine up to at least 300 lbs" but really it's hard to say how heavy is too heavy for a bike to carry (even the posted limits on bikes that have them are somewhat arbitrary, mostly to please insurance and warranty issues), since your riding style, terrain and sometimes luck will have a lot to do with how much strain you put on a bike. For example, a 300 lbs rider might ride a folding bike that has a much lower suggested rider weight for years without issue, but if he should bend the seatpost by hitting a pothole, the manufacturer can refuse to offer a warranty replacement.
Generally though, big riders can ride the same bikes as anyone else, but can also expect to wear stuff out faster. Tires, chains and brakes pads are all items that wear over time, and the bigger you are, the fewer miles you can expect to get out of them.
Super-lightweight parts are best avoided (if you're commuting, avoid them anyway), and if there's one part of the bike I would recommend a heavier rider put extra attention and money into, it would have to be the wheelset.
Wheels suitable for a heavy rider are not necessarily the most expensive ones out there (those are often super lightweight and designed for racing, therefore a poor choice) nor are they the cheapest ones. Most entry-level mountain or commuter bikes, for example, come with inexpensive wheels with single-wall aluminium rims. These are adequate for the average rider, but will be far more likely to suffer issues with a heavier rider. Instead, look for wheels the next step up with double-walled rims. These will be much stronger, stay in true better and have fewer broken spokes. (I have a photo around here somewhere illustrating the difference between double- and single-walled rims, I'll upload it when I can find it, but the major thing is that a single-walled rim has a "U" shaped cross section, but the double-wall adds another piece of metal, like a cross-brace across the "U", making for a stronger, stiffer rim).
Wheels with lower-spoke counts are also not worth a big rider's time. 32-spoke or 36-spoke wheels are the way to go. One innovation from the racing world that is worth considering are deeper-cross-section rims. These rims are designed to be stiff and strong with 16 or 24 spokes, if they're built up with 32 or 36 spokes they are pretty bombproof (you may have trouble finding stock wheelsets built up this way, I build a lot of my own wheels specifically so I can get the combination I want). Wheels made for touring bikes or tandems are good off-the-shelf options for road bikes, while heavy-duty mountain bike wheels are not hard to find at all at a reasonable price.
Fatter tires are also an asset to the heavy rider. More air volume means more cushion and less likelihood of a pinch flat. Most mountain bike tires are fat enough, but big riders should make sure they're always pumped towards the higher end of their pressure range. On a road bike, or bike with 700c wheels, 32-38mm tires are great. Many road bikes won't fit more than a 25 tire width, in which case, go with the biggest you can fit, and pump it up to the max recommended pressure.
On other parts of the bike, use common sense. Replace plastic pedals with metal, and avoid saddles that are too squishy (you'll bottom out). Suspension forks may cause problems for the big rider, if they're non-adjustable, expect them to be too soft and prone to bottoming out just from your seated weight. If they are adjustable, their highest tension/pressure setting may or may not be stiff enough for the best ride quality.. Rear suspension bikes are best avoided entirely unless your willing to spend well over $2,000. Overall, the simpler the bike is mechanically, the more likely you are to be happy with it.