(and some chemistry thrown in)
I'll explain the really simple differences between red and white, then I'll give you a load of technical stuff.
All grape skins and seeds contain "tannins", but the skins of red grapes contain more tannin than whites, and these tannins are also coloured. Obviously a red grape usually makes a red wine, but it is still possible to make them into a white wine (though not the other way around!). For example, most sparkling wines have Pinot Noir in them. We can do this because very few red grapes that have coloured juice - most grape juice is almost white. (For the nerds, a variety that has red juice is called a teinturier, meaning it holds colour).
Turning to wine, reds have tannin, whites don't (have much, usually). Tannin gives wine colour, texture, and can make a wine bitter. Tannins are only in the skins and the seeds, so to make a white wine we press the grapes as soon as possible, giving nice clean juice. If I want to make a red wine, then I keep everything together as long as possible, to extract colour and texture. That's pretty much it.
What are tannins?
The definition of a tannin is something that precipitates protein, and the source of the word comes from the Latin word for oak bark (tannum), which gave its name to the process of tanning leather. Tannins are widespread in nature, produced by many plants. You probably (unknowingly) eat or drink a lot of tannins, and the most obvious example is tea. Tea gets its colour and a lot of its character from tannin.
Tannins are contained within the complex family of phenolic compounds. That doesn't sound like a nice thing, but they are actually vital to life. Many phenolics are natural anti-oxidants, protecting the grape against things like UV in sunlight UV, or diseases like mildews. Red grapes produce much more of these in their skins than white grapes, and they act like a natural sunscreen. Though I'm not sure whether that is applicable in the photo below.
They are of primary interest in wine because they affect what we call "mouthfeel". It is a technical sounding term for how the wine feels in the mouth - is it rough, does it grab you at the front or the back, or all the way through. This is "astringency". They can also affect a wine's bitterness. What is strange is that even though we think we know how tannins work, we can't explain why different tannins give different sensations in the mouth. Tannins also give wine colour. An example of a pigmented tannin is shown below. It is the sort of thing we use to scare baby winemakers into paying attention during class.
The secondary interest is because it is claimed that the tannins in red wines contribute to positive health impacts. However, the research on this is highly contradictory, and no one is really sure whether they actually work or not. However, for now I think I'd prefer to stick with the side that says they do work...
Using winemaking to change wine style
Tannins are a major part of red wines, but we can change the style of the wine by changing winemaking practices. For example, if I want a big red, I may pre-soak the grapes for a while, and then soak for a while after fermentation. During the fermentation, the carbon dioxide released by the yeast makes the skins float to the top, so we have to mix the skins back into the wine to get the tannins out. I can vary the amount of mixing to get different levels of extraction. I could also try different temperatures - higher temperatures seem to get more extraction, but significantly speed up how fast the yeast work.
To make a lighter style of wine I could reduce the length of time I ferment everything together, or I can use a grape with lower levels of tannin. Grapes grown in cooler climates tend to have lower levels of tannins than warm-climate grapes, so this is another way to change the wine style.
Finally, close to bottling I can tweak the tannins using fining agents such as egg white (remember that tannins precipitate proteins - we are just using a protein to precipitate the tannin). This softens the wine and makes it more approachable, especially when young.
Tannins in white wines
In recent years, there has been much more interest in the role of tannins in white wines. The skins of white grapes do have tannins, but many are different to red grapes, and often at lower levels. Even white wines have some tannins, and there is much discussion about the roles that they play. When talking about white wines however, instead of saying tannins, we say phenolics (even though tannins are phenolics)
Generally when making white wines we want to avoid high levels of tannins, and we add fining agents to the juice to remove as much as possible. But fining agents can reduce flavour if used at high levels, so we usually split press cycles into "free run" and "pressings". The free run is an early fraction that doesn't need much squeezing, and contains low levels of tannins. Then we increase how much we mix up the skins, and how hard we squeeze - this is called the pressings, and has more tannin. We can then add more fining agent to the pressings - thus protecting more of the flavours in the free run.
Sometimes we want to make a wine that has a lot of palate texture, and we keep some or all of the pressings. The Reserve Chardonnay gets a little pressings, the Pinot Gris gets everything. Over time these tannins add roundness, and make the wine really good with food.
What else is different?
As a rule, all of our red wines are kept in the winery for a number of months before bottling, whereas most of our white wines are not. I go into this in more detail later, but the process is called maturation. The less tannin a wine has, the less maturation is required before the wine can be bottled. Essentially this is just a chance for things to settle down a bit. The lighter, fresher whites can be bottled quickly, to retain that freshness. Our Reserve Chardonnay gets up to 12 months in barrels to allow the oak to integrate, and to develop extra texture from time in contact with the yeast. The maturation of the reds depends on the wine - a lighter, fruity style will be given about 6 months in old oak, whereas a more tannic wine such as Cabernet Sauvignon is given up to 12 months maturation before bottling. This will be covered in much more detail in the section about wine maturation.
As you can see, when it comes to grapes, colour is usually only skin deep, however those thin skins can have a huge effect on the finished wine, giving red wines the colour and mouthfeel that make them so different from white wines. The difference comes about from a combination of the grape and the winemaking techniques.